“Do we know how the court is defining ‘biological sex’? Genetically? Chromosomally? Hormonally? Aesthetically?”
If you really want to raise your head above the parapet and answer that, then the SC cited a case from 1970 called Corbett v Corbett (otherwise Ashley).
A man had married a post-op transwoman in 1963. They then split up 14 days later.
The transwoman claimed for maintenance but the man sought an annulment of the marriage (ie saying that the marriage had never happened) because, at that time, a marriage must be between a man and a woman.
The court ruled that the transwoman was a man and that sex at birth cannot be changed.
The court went on to say that determining sex was down to just three things:
(i) Chromosomal factors.
(ii) Gonadal factors (ie presence or absence of testes or ovaries).
(iii) Genital factors (including internal sex organs).
and nothing else, so not hormonal factors or psychological factors.
This case has been cited with approval on a number of occasions over the last 50 years. It is pretty much well accepted.
This is why the SC said at para 171:
"Although the word “biological” does not appear in this definition, the ordinary meaning of those plain and unambiguous words corresponds with the biological characteristics that make an individual a man or a woman. These are assumed to be self-explanatory and to require no further explanation."