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This way it's unambiguous.
I don't think we need "physical training", which is "old-fashioned" according to Collins.

I agree with Opencooper. (having gloss type=AmE and BritE etc. like I've previously suggested would maybe "non-international" glosses less 
controversial)

Sorry for belaboring this, but this has important implications for future glossing, so I feel it needs to 
be. If such argument is better suited for the mailing list, I can take it there.

The editorial policy does say to "make the translations as international as possible", using the same 
example you gave. However, I always interpreted this as referring to the primary gloss. Indeed, our entry 
for 「大学」 has both "university" and "college" in that order. Going through the above lexico.com link, tons 
of our entries include both English variants of terms (arbitrary examples: 駐車場, 薬局, ナス). Even for 
katakana entries with a specific lsrc, we list the variants (e.g. クッキー, ラップ, ズッキーニ).

Including English dialectal variants is not an unusual practice, especially among the Japanese–English 
dictionaries we consult such as Chuujiten, where there'll be annotations such as 《米》 and 《英》. We can't 
just service an elusive "international English", but should support multiple English renderings. In fact, we 
add glosses all the time when they are useful for reverse lookups.

Finally, "gym" in this usage, while possibly unfamiliar to non-Americans, isn't uncommon. See above 
references. Unfortunately, there will be many cases where polysemes might cause confusion, but this is not a 
problem when there are other glosses to clarify and it's not leading.

Yep, but it doesn't travel that well. I've always tried to keep the translations as universal as possible, e.g. using "university" rather than "college" which is primarily a US term. "gym" in BrE  has a lot of different associations, and isn't usually a portmanteau term for physical education.

Strange, seems OED's lexico site doesn't let you link to the US version, but the sense I meant was:

    2 Physical education.
    ‘I'm taking just one more semester of gym’

As in "gym class". I can verify that this is what we called PE in the states, and there weren't any 
gymnastics involved :D.