I've lived in most of the places mentioned at some point or other.
A lot of it will come down to schools. I'm in Horsforth now and a lot of people have moved here for the schools. Main thing being that both the primaries and secondary are well regarded. (I know people whose children go to Lawnswood, Ralph Thoresby too and they are happy with them so I don't think it's as big a difference as is sometimes made out). I personally like the location of Horsforth, and being on the train line, even with Northern Rail being as terrible as they are, is so much better than having to rely on a bus. Yes, car parking at the stations isn't great, but from most of horsforth you're not more than about 15 minute walk from either. I personally don't mind the airplane noise, and never really notice it indoors - only when out in the garden, but even then there aren't that many planes.
Meanwood is great, the primary there really is one of the best anywhere. Location wise it really is very quick to get into the city centre, there's a nice park, things going on, still close to Headingley.
Weetwood is the other side of the park from Meanwood and would maybe have the sort of house that you are looking for. Again, the local primary there is very good.
Unfortunately all the well regarded primaries are likely to be oversubscribed. I do know quite a few people who have managed to move schools after a year or two though. So it's not necessarily the case that you'll be stuck with a school you're not keen on forever.