Strength / Weight Training

Totally your call but I'd go with some trainers with a solid sole and learn to squat on those.
Recently bought some Adidas crossfit type trainers (in hot pink, @Chalfie would approve) and they are great for everything in the gym.

I'd say try to just find what works for you. Try squatting in flat shoes (converse / vans / plimsoles / similar) and see how it feels. If you don't feel the need to get WL shoes then there isn't one.

also see folks use little plates (2.5-5kg) under their heels to get a bit of lift in normal shoes - though not tried it myself so can't speak to whether it caused any stability woes

I started with converse, which are good, but then I didn't do much for ages, came back and ankle mobility was more of an issue so got shoes.

On the shoes thing, I tend to just kick off my trainers and do squats / DLs etc in my socks...

Has the added bonus of helping keep people from lingering too near me or the squat rack.