How does a person know that he is a prophet?
No one makes himself a prophet. God chooses him and speaks to him so clearly and so often that no doubt remains, and God confirms him through signs that others can see as well.
Prophethood is not a career a person chooses, nor a feeling he talks himself into. It always begins with God. He chooses a person and speaks to him. The first verse above names the ways this happens, namely revelation, a word from behind a veil, or an angel sent as a messenger.
This speech of God is different from one's own thoughts or wishes. It comes with power, often unexpectedly, and it carries its own certainty. The account of the first revelation shows that even the Holy Prophet Muhammad (saw) was shaken at first. He had not sought it. Only as God spoke to him again and again did the shock turn into firm certainty.
There is also a sign that others can test. God tells His messenger things from the unseen, and they come true. He protects him and makes his message grow, even against great opposition. Someone who merely imagines it cannot fulfil such promises. Time exposes him.
So if you ask how to recognise a prophet, do not look at grand words. Look at the life of that person, at his truthfulness even before his claim, and at what God brings about through him. Someone who is not a prophet does not find this certainty within himself, however devout he is and however much he serves others. It is a gift, not an achievement.