Originally, the meaning of a remix is when you take the original recording and redo the mix. Mix being not only the levels and EQs on the console, but also in which order some parts of the songs are presented, extended, cut short etc. Basically, what the mix engineer does. This is true to this day, but technology has changed so much that you have to factor in sampling and whatnot.
A cover is simply playing the song as it was originally made. You don't need to attempt to sound like the original, but you are keeping the structure as it is. Basically a cover is when you get the sheet music version, and play it. So you can make a guitar cover of a piano song as long as you keep the chord progression and structure the same way (meaning you CAN make an electronic cover of a song.) Tempo is usually irrelevant because sheet music doesn't note tempo precisely down to the beats per minute.
For a lot of music for which we don't have the original recording (like a lot of classical music) we usually call them performances, or interpretation.
For music that you keep the basic original structure and feel but to which you add/remove layers of instrumentations to, we usually refer to those as 'arrangements'
What a lot of people do today however, cutting up a snippet from a song and use it as the hook in another completely different song has been given the name of 'vignette' in a lot of circles, 'mash-up' in others...mostly when more than one sources have been used.
TL;DR: Just credit the original artist and stop caring about this, it's irrelevant