afternoon
sounds like sensible advice and like it'll be perfectly safe.
I think that they'll probably give you a long list of things you can't do, but probably not too many ideas of things that you can continue to do (esp abs exercises or exercises to help the muscles you'll use in labour). you'd get more out of it if there was someone willing to give you some more positive ideas about what you can and can't do and they'll err heavily on the side of caution, but you'll be v. safe. I'd try and collar the trainer he mentioned as she'll probably be interested to help you.
the balance thing/swiss ball is really a safety issue for them - I get my clients to do lots of work on the swiss ball but I'm there one-to-one to make sure they don't wobble off and land on their bum! It's all the more important really to be using the abs for balance etc when you're pregnant. It's not disallowed for any physiological reason if you see what I mean.
spin/rpm is ok but as you get bigger you'll probably find the compression on your bump uncomfortable (from leaning over the bike)and you'll also find that the pressure on your pelvic bones may be uncomfortable as the ligaments soften. Careful of your balance changing getting on and off the bike. Make sure you get your saddle height right. Don't go at usual spin pace - if you think about a scale of 1/10 with 1 sitting on the sofa eating pie and 10 sprinting for your life, you shouldn't be going beyond a 7/10. (roughly). You should be able to talk - it's ok to get a bit puffed but you still need to be able to string a sentence together albeit in short bursts. If you feel knackered, stop. Drink plenty of water and don't get too hot. Eat regularly. Now I sound like my mother!
pilates is fine as long as the instructor understands what you can and can't do which I'm sure they do. If you're unsure about anything, don't be afraid to ask if it's ok and if anything feels weird or just 'uncomfortable' this is a good sign to follow - stop and let them know.
You might hear some very old fashioned advice like no abdominals work/no lunges/no squats/no lifting weights - none of which is true but at least you'll be safe!
Stick with it, I know I'm biased but being fit while pregnant is so important - for your sanity, health and for the baby. Shout if you've got any questions, like I've said I'm a geek when it comes to all of this and love nothing more than gassing about it. I'm giving up my one-to-one clients in the next couple of weeks (am getting v. pregnant now) so have lots of time to bore everyone with exercise advice!