Yes, it's ok to say "I hurt". For whatever reason, "I hurt" might be used to describe a general, often non-physical, non-life-threatening pain. Ngram suggests the use of "I hurt" has grown hugely since 1970, so my guess is that it became a common phrase in self-help and self-awareness books, websites, and other media, and nowadays has spilled over into general usage.
The present progressive is also acceptable (and suggests more urgency):
I'm in pain!
As in your first example, a common way to refer to pain is by mentioning the source of the pain:
This really hurts!
This is really painful!
Or more colloquially:
This hurts like a sonofabitch!
There are, of course, many more colorful ways to express pain.