Afraid I can't speak from personal experience, but I'd say that they're YOUR therapist and there to help YOU. So if you're not comfortable talking about something then absolutely they should respect your wishes. They need to work with you at YOUR pace. Therapy shouldn't just be a "textbook procedure" it should be personalised to you.
True, they may encourage you to push yourself a little at times through your therapy but they should ensure that you're not pressured and feel safe in pushing yourself, and that you are ready to/want to push yourself.
It might be good to give them just a outline at the beginning though (if you feel OK with this) without any details if you want. Something as "simple" as telling them it happened but you don't want to go into any detail at the time. That way if they ask you questions or make suggestions they can at least be sensitive around what they're asking/suggesting. And then as sessions go on you may feel more able to give more details.
But at least it's good that you're not still seeing the last therapist, just make sure you're expecting a lot more than that from your current/next one.
Best wishes
Alison