I believe Iffley Meadows, as a SSSI, extends further up. I would keep the previous boundary, since it seems more legally defined.
Longbridges Nature Park would then be only the northern end of the island (where currently there is a duplicate park/reserve).
Let me know if you agree.

In fact it is locked, with codes given to members. https://www.ouwg.org.uk/2020/04/06/covid-19-advice-for-members/

I don't know whether they are ever actually locked, possibly relying on trust only.
All the best.

It is technically open to members only, with a £5/y subscription. See: https://www.ouwg.org.uk/visiting-boundary-brook-nature-park/

Sure. I ended up with the Broad-Magdalen corner which seemed more central from a geographic point of view, but Carfax makes perfect sense. Go ahead!

Thanks, you're probably right. OS OpenData has it as "Lords's" however. I can also find a couple of uses in published work both for "Lord's" and for "Lords". I will double-check the local sign next time I'm there.