solipsism holds that there is no knowledge of anything outside one's own mind
It is the definition used by many philosophers, but it is extremely misleading for we may indeed believe that we don't know anything outside our own mind and yet believe that there are other minds.
Plus, this is the default position, not the weird or extreme philosophical position suggested by the term 'solipsism'.
The only reasonable definition of solipsism is that it is the belief that your mind is the only one. We normally believe that there are other minds since we don't have the choice--it is apparently hardwired in to our brain--,so it makes sense to ask what would happen if we could really believe that our mind is alone. Interesting question, but no one really believes that their mind is the only one outside pathological cases. Arguing for the sake of argument that other minds do not exist is just silly. You are obviously not going to convince other minds, and you also obviously don't need to make your argument public to convince yourself.
if one cannot prove that other minds exist for sure, does this imply you are a solipsist?
No.
Proving a conclusion does not guaranty that it is true. It just proves that if the premises are true, then the conclusion is true. And we usually don't know whether our premises are true, so we don't know whether our "proven" conclusion is true.
Plus, that you cannot prove a conclusion doesn't mean that you believe that it is false. Ask mathematicians about conjectures.