I've been trying to capture a fourth Sylesti. However, when I catch one, it doesn't go to my stable, despite the fact that there are spaces in it. It seems to have been released, despite the fact that it was certainly captured. What am I doing wrong? Do I have to do something before I can catch a fourth Sylesti?

There is a chance that you scare the sylesti away if the trap fail on it, if you don't read the log you will not be able to know if you realy caught the pet or not as there is no difference in the victory screen.

Given your level, I would assume you are using the lowest level traps. These traps are frankly a complete waste of money. 99% of the time they fail and the pet flees.

If you are looking for another pet you are much better off using the advanced search (blue writing, top left). You will save money and you will definitely get a pet out of it. Plus you can pick traits you like.