How do I ensure I'll be able to claim authorship over a math proof, as an amateur math student?
Moreover, it would be really embarrassing to me to talk about this with people I know, especially if they rapidly find a mistake in the proof. I fear they'd think I'm silly for even considering the possibility I may have proven something no mathematician has been able to prove before

My advice for this concern is to propose your work as an attempt rather than conclusive. It is normal and okay for attempts to fail: most do. If you reduce your expectations publicly to those that you hold privately, there is little cost to failure and substantial benefit to success. Ask others to find errors in your work, and be humble.
If they can find no errors, they will be sufficiently impressed. If they do find errors, you can be appreciative and they will feel useful for having found and taught you something and you need not be embarrassed, only thankful.