Indeed, M-W is little used in the UK and not even particularly widely known here - whereas Oxford is known worldwide.
Incidentally, Oxford's definition of "transwoman" is "a male-to-female transsexual" ( www.lexico.com/definition/transwoman ). This is a somewhat more satisfactory and objective definition.
"Transwoman" is a propagandist term aimed at promulgating the idea that M-W endorse here. After all, usually if a word ends in "woman" then it names a type of women. "Transwomen are women" sounds much like a tautology than a controversial statement, and yet many older people probably don't even know what a transwoman is (for a long time I thought it meant F-to-M).
With the Oxford definition, you can see that TWAW translates to "male-to-female transsexuals are women", and the controversial nature of the statement, and the fact that one ought to have the right to debate it without being considered hateful, become so very clear.