They are correct.
But they carry a lot of emotional and dramatic weight because of the "I am the man who" structure.
Scenario 1: I'm in an elevator, the power goes out for 5 minutes, I use the emergency phone to talk to the building manager, and then everything is ok. When someone mentions it the next week, if I say "I am the man who was in the elevator", I would sound pompous and ridiculous. The more normal expression would be "That was me."
Scenario 2: I'm an assistant on a dig in the Great Pyramid, and am the first person to see a relic that eventually proves something momentous about the ancient Egyptians. 30 years later, I overhear someone disputing that fact. I respond by saying "I am the man who first uncovered the whatsis of Ramses: you, sir, are wrong!" In that case, "I am the man" would be appropriate.
Your added example of "I am a man who..." is less dramatic, although still not idiomatic. People are more likely to say "I like to travel" or "I'm the kind of person who likes to travel".