The Wikipedia article sums up pretty close to what you have described hence I would vote for `variation`. 

Here are a few examples (Not classical or mainstream jazz since they are completely based on variations anyhow) 

Taking a cue from one of your sentences, I would recommend [this video][1] from the comedy-rock band *Axis of Awesome* (mostly known for their [4 chord song][2]) mocking the key shift usually appears in commercial pop songs (the relevant part starts from 2:35). 

Another nontrivial example (that I can think of right away and that would make an example in this context without being too complicated) is from the song ["Learning to Live"][3] out of Dream Theater's Images and Words album. Around 7:40 the keyboard sets up a riff and then cues for a slight increase of tension and together with the lead guitar, perform a long `variation` around this theme up to the guitar solo and return to a similar pattern until the vocals come in.  

And of course my personal favorite is Vinnie Colaiuta's (or the band *Karizma*'s) song "I'm Tweaked". The song is based on disturbing the listener's 4/4 beat feel here and there with having some instruments playing straight 4/4 but in the meantime having the drums occasionally shift back and forth a 16th note.


  [1]: http://www.youtube.com/watch?v=fSuIZ11JdUg
  [2]: http://www.youtube.com/watch?v=5pidokakU4I
  [3]: http://www.youtube.com/watch?v=qY0-k4uaKKA