Do not sign anything from your current employer for any reason. You’re doing well there.

Block them all on LinkedIn and social media.

Do not share any further information, even “not the same industry”, with them.  “That’s none of your business” is answer enough, and they have been unprofessional enough you’d be justified in not serving the rest of your notice IMO.

Non compete clauses are only very narrowly enforceable in Canada (https://www.canadaemploymenthumanrightslaw.com/2018/02/employees-non-compete-agreement-enforceable/) and their claim would not stand up.  However, you are right to avoid conflict by not updating your LinkedIn for a while (there’s no reason to do it quickly really). If eventually they find out it is super unlikely they’ll really do anything and usually your new company will be happy to send them a lawyer letter telling them to get bent.