Agreed, the UK and the rest of the world could be deemed as close or far depending on your perspective, but were here to try and clear up a few things - though not all of them as we discover some anomalies every week...but for 99% of everything, here goes...are you sitting comfortably?
For what it's worth I'll explain to you how these things appear to me:
My approach to any such question is to take it outside the immediate, local, context which gave rise to it, then try to look objectively at what the reasons for any development might have been - or how might history have affected the development.
My belief is that, habitually, the width of the gouge flute was used to describe the size of a gouge. It makes reasonable sense in a world where gouges were hand forged, and recognises the fact that the flute size is more important, functionally, than the outside diameter. In fact, in former times where all gouges were forged, the outer diameter was probably beyond easy recognition since the gouge was unlikely to have been forged from a bar of accurately drawn steel stock bar. By contrast, the flute was the part that the forger would concentrate on making of the correct size and constant cross-section. The flute size determines the 'cutting capacity' of a gouge, and also the radius of the edge presented to the timber (and hence the 'smoothness' of the cut surface created - the larger the radius/diameter, the smoother the surface) and is likely to be the factor that would be of interest to the turner. Bowl gouges in the specific form in which we use them today are a recent development (ca. 50 years or less) and historically most gouges were of the type we would today classify as of the 'spindle' form. Bowls can be made entirely using scraping style tools (as many more restricted region of the world still do) and even using such tools (despite what you might read in the turning press) a decent finish can be achieved.
As a matter of interest, I know one apprentice-trained 'pro' turner who tells me that when he started his apprenticeship, he was supplied not with tools, but with tool steel bars, from which he was required to fashion his own tools. A Turkish turner whom my father met in Istanbul a few years ago was bemoaning the fact that he could not obtain modern HSS tools: instead, he bought HSS bars and ground out the flutes, as well as the cutting edge shape, by hand using an angle grinder, before hardening and tempering them, also by hand and eye, over a charcoal hearth. The tools he produced were indistinguishable from factory production!
Why are bowl and spindle gouges measured differently? In fact, they're not! They can both be seen as being the width of the flute. The difference is in the positioning and size of the flutes in modern gouges. The flutes are produced using a milling cutter. This is uniform in cross-section shape so gives a consistent curve. The flute in spindle gouge takes up the full width of the blade material - in creating the flute, one half of the depth of the blade material is removed - and the flute is perforce the same width as the bar material from which the blade is made. The flute needs to be as large a radius as possible to meet the requirements for a smoothest possible (ripple-free) cut.
In a bowl gouge we need to have concern for the strength of the tool blade when subject to bending forces as we work deeply inside our bowl. We also have less care about the detail finish resulting from a deep cut made with a narrower edge, since we don't take proportionately deep cuts with a bowl gouge. To maximise the bending strength of the blade we need to maximise the bending section - keeping as much material as we can at the greatest distance from the bending plane (neutral axis) of the cross-section. Basically that means retaining as much material as possible at top and bottom of the bar. However we need a flute so the best compromise for these two requirements is to keep the flute relatively narrow and to incise it into the top of the bar, without letting it break out at the sides. Hence we find we have created the deep fluted gouge, known to most modern turners. How wide is the flute? Well, it's a little narrower than the diameter (width) of the blade steel. How much narrower? - well, abut 1/8 inch - i.e. 1/16 inch (1.6mm) each side of the flute. If you make the flute much wider then that, the side walls, being less than 1/16 inch in such a case, would retain little strength. Hence we end up with the flute width being about 1/8" less than the bar diameter of a bowl gouge. The only place where you can consistently measure the width to be able to compare tools or to give a constant definition, is at the top of the flute, where it intersects the outside diameter of the bar stock that is the blade. The relationship is fairly constant regardless of the bar diameter because we are talking about physical limits of machining, minimum required bar strengths for the gouges and acceptable flute profiles (to permit shavings to flow clear of the flute).
You will know that if you took critical measurements of flute width to bar diameter for a range of bowl gouges of different flute profiles, by different manufacturers, you would get different values for each and every one. But they would not be wildly disparate. Sufficiently close, in fact, that as a 'shorthand' for referring to bowl gouges it is remarkably convenient, pays its respects to tradition, is easy to use (once understood) and would be perfectly satisfactory (if somewhat irrational to the uninitiated) were it not for the penchant of manufacturers to either misunderstand or to try to be 'clever'. (Viz. the HS1 gouge from Henry Taylor!!) SEE FOOTNOTE
So, I believe, the fact is that the process started 'the other way around'. The flute defined the gouge; the size of the flute discriminated between gouges; the method of producing gouges changed when we started to make them from bar stock, but the flute remained an important dimension to the users of the time; to produce a gouge with a flute dimensioned as before, the material size was increased above the nominal flute width; the increase in size necessary was 'reasonably' constant so one could equally use the flute size or bar stock to define the size. A unitary system is better than a binary one so we used the same definition for both styles of gouge (as they developed) and "we all knew" that a bowl gouge was always made from steel 1/8 inch larger diameter than the nominal tool (flute) size, whilst a spindle gouge was made of steel the diameter of the finished flute.
As long as 'all' turners were well-trained, narrow-minded and working hand in glove with the manufacturers, the system was fine. This is especially true since most of the old turners would see the flute dimension as the important factor, not the bar diameter. Once we got to the post-war (WWII) explosion in recreational turning, none of that held true any longer. Most recreational turners are far from professional or have historic knowledge in depth so unless there is an easy logic to the nomenclature, it simply confuses and confounds - and causes errors to be made. And, clearly, the manufacturers do nothing to help the situation, tending to add to the confusion rather than clarity!
For my part, I would prefer that, for the sake of modern turners using modern tools, we simply referred to the bar size used for the tool - what some of us have come to term 'The American System'. It is easy to understand, requires no interpretation, and can be easily explained. It should also enable gouges to be referred to without any risk of confusion. However it would require the removal of that huge roadblock called 'tradition' and this is something that we as an industry have become very proud of, rightly or wrongly. I am sure it will simply be one more step along the way and new pitfalls will lie ahead - after all, we know how smooth thread standardisation was...don’t we...?
So that’s it, the world of gouge measurements as we see it. The minefield of flute vs bar measurement. A simple rule of thumb - ‘traditional’ manufacturers (e.g. Robert Sorby, Henry Taylor, Hamlet, Crown Tools etc...) will refer to flute size, i.e. a 1/2” bowl gouge, a tool which the modern manufacturers (e.g. Carter & Son Toolworks) and market entrants would call a 5/8” bowl gouge, the stock bar diameter. But yet again, even Carter have some tools which blur the lines in their Mahoney range, because Mike had his terminology which matched the traditional approach...sigh.
Well done if you’ve made it this far, might I suggest you go and put the kettle on again as that mug in your hand is surely cold by now.
FOOTNOTE - for those who made it this far...
The significance of the HS1 is that, from its introduction, it was described as being a 5/8 inch bowl gouge. However it was made from 5/8 inch bar stock and thus, using the traditional British nomenclature, 'should' have been described as a 1/2 inch bowl gouge. If you look at the Bowl Gouges page of our own website you will see that it is referenced as being a 5/8 inch gouge. We were left with a quandary when we first listed it: if we called it a 1/2 inch gouges we would be in conflict with the manufacturer's description whereas if we called it a 5/8 inch gouge we might find customers expecting a larger diameter bowl gouge than they received. One cannot win this argument: trying to explain the fact if a few words on a website leads you into ever deeper quagmires of confusion. Eventually I used my powers of persuasion on the more recent owner of Henry Taylor to get them to change their description. Thus if you were party to their factory price list you would see that the tool is now described as being a 1/2 inch gouge - which in conventional terms is what it is. However the only Taylor catalogue in circulation is an old one and that still uses the original terminology. Maybe they expected to flood the US market with this new 'wonder tool' of its day and therefore decided to use US terminology - and make life harder for the non-US markets. More likely they simply thought that 'bigger is better' and didn't really give it much analytical thought at all. So that is why I 'point the finger' in isolation at the HS1: it is a glaring example of what happens if you do not follow convention, and of the confusion that is then caused - great bowl gouge though!
Without wanting to make your learning yet more difficult (but yes, it’s going to happen), I would also advise you that many modern tools have dimensions very different from either their nominal or stated 'bar size' dimensions simply because inch size bar stock is no longer widely available, metric size stock does not enjoy consistent availability, and re-grinding stock to nominal standard sizes prior to manufacture of the tools is uneconomic. Hence you maybe find that your 3/8 inch bowl gouge would not measure the expected 1/2 inch diameter (12.7mm) but may be 12.5mm; 12.7mm; 13mm and right up to around 13.5mm (because of bar stock manufacturing tolerances). And the situation is likely to change with each delivery of raw material. This not only makes bar stock measurement systems challenging, but that multi handle you bought...there’s no guarantees that your 1/2” is the same as someone else’s idea of 1/2”.
Who'd want to be a tool manufacturer, eh?
There’s a good chance I might need something a little stronger than tea after all that...