The rule for a OWE stopover reads:

152N . 1. A STOPOVER IS A BREAK OF JOURNEY OVER 24 HOURS.

My planned flight arrives at LHR at 21:30, and I want to connect with another that departs LHR at 21:30 the following day.

The way I see it that's 24 hours exactly (not over) so it should be allowed as a transit. Will the airlines agree?

I guess that depends on how persuasive you are. I agree with your interpretation though, and armed with the star file and some gentle brow-beating, you should be able to prevail. I have no direct experience of this myself though.

It's a CONNECTION. If your departing flight is at 21.31, then it's a stopover.

When I asked AA how long a NY connection could be (LHR-JFK-GIG), they told me that they count anything over 23 hours 59 minutes as a stopover, which incurs an extra charge depending on the fare. I'd advise you to call the airline to doublecheck, but I'm sure you could persuade them that one minute (21:30 landing one day and 21:30 takeoff the next=24 hours unfortunately) between friends isn't a big deal. Then again, it might be a computer thing which a nice agent would have to manually override. I'd be curious to get their answer.

I've no experience of how AA deals with this but if they treat it as "black and white" (i.e., 23.59 or, for that matter 24.00 hours and not a minute more) then they're inconsistent with how they treat domestic 'sticker' upgrades. Those domestic 'stickers' are good for 500 miles but AA gives you 50 miles leeway.... so for example TPA-LGA is 1,012 miles but only requires 2 500-mile stickers to upgrade, i.e., anything up to 1,050 miles only requires 2 stickers.... 1,051 requires 3. Admittedly, stickers only come in 500-mile chunks whereas hours come in minute increments (at least as far as schedules are concerned) so the analogy isn't perfect. Ultimately, there has to be a "black and white line" somewhere but to insist that it's at 23.59 seems harsh (and perhaps simply wrong, depending on which wording you're reading). However, if the wording is ".. depart within 24 hours ..." then 'within' would imply 23.59.....

AA as in the AA rep.. a person, who has no clue but goes strictly by the rules.

Hmm. Funny that the SABRE computer treats it as a connection. So I would go by what the computer says!

I don't exactly know what flights the OP is taking.. but here is an example of what SABRE or any other CRS will do...

AA 142J 08SEP 3 JFKLHR SS1 0830 2015
AF2271C 09SEP 4 LHRCDG SS1 2015 2225

You see the fare calculation below..... x/ means it's counted as CONNECTION.
NYC AA X/LON Q8.00 AF PAR