@ChattyFibre,
Counsellors should not be telling anyone else (that you do not agree to) why you are seeing them.
You may not need to meet a counsellor on school property or during school time.
Also, a good counsellor can help you figure out what you do want to say to anyone who might find out you are seeing them.
It's definitely not easy starting a conversation about needing help, but you've taken a step by at least telling strangers you can't see.