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 from the comedy-rock band Axis of Awesome (mostly known for their 4 chord song) 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" 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.

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 from the comedy-rock band Axis of Awesome (mostly known for their 4 chord song) 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" 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.