I'm proud of something imply
I'm taking credit for something? I just observed that the former phrase only works with my own accomplishments or with related people, but not with unrelated people, e.g.
I'm proud of Tom's result works only if I have some kind of relationship with Tom, like I'm his father or teacher or trainer.
No, you cannot imply this.
Being proud of something implies that you or someone you associate with has accomplished that something, and that the accomplishment gives you satisfaction or happiness.
Taking credit for something means that you are somehow responsible for that something - it wouldn't have happened without you.
You can be proud of your son for winning a race, it does not mean that you are taking credit for it. You can also take credit for something without necessarily being proud of having done it. For instance someone could take credit and get a raise for reducing expenditure in their company but doesn't feel particularly proud because to do it they had to fire some people.