"Re" Words

I wondered out loud-
"What are we 're-creating' when we participate in recreation?"
My wise Auntie responded, "Well, you're certainly recreating muscle tissue. You may also be de-stressing and recreating mental sanity and calm."
Made sense.

So now Anthony Weiner has resigned.
"Re-signed" what?
How does quitting equate with signing for another hitch at something?

I think I "re-sent" that use of the word.