Is there a word for "piano reinterpretation"?

I really love the idea of people taking a simple melody, perhaps a well known one, and making their own piano interpretation of it, implementing some parts of the original and adding their own bit, many times in free-form. I would like to be able to search for these, but I can't seem to think of a single word to use.

(Sorry if this was posted out of place, but I think "English" would be closer than "Music" in this case)

Rough examples:

BBNG - Bastard/Lemonade

Max Payne 3's menu theme has two movements that play around with the original idea

Is "piano reinterpretation" really the simplest term to use?

Solution 1:

Speaking as someone who has had a musical training, I think you may mean

piano arrangement

arrangement

3A musical composition arranged for performance with instruments or voices differing from those originally specified: ‘Mozart’s symphonies in arrangements for cello and piano’ http://oxforddictionaries.com

Example

New York-based AlfieProducer plays a piano arrangement of 42 popular songs like Robin Thicke’s “Blurred Lines,” Daft Punk’s “Get Lucky,” and “Radioactive” by Imagine Dragons in this impressive video. http://laughingsquid.com/a-piano-arrangement-of-42-pop-songs-like-blurred-lines-get-lucky-and-more/

This term is used for most cases. The main exception occurs when most of the piece is extemporized in which case it might be called an improvisation on a theme, e.g. Improvisation On A Theme (Pachelbel's Canon In D Major) performed by David Lanz https://www.youtube.com/watch?v=qqGrB4MxdsY

In the classical world you will also hear the term variations on a theme. Usually a well-known melody is played followed by many different versions of it - often virtuosic in nature, e.g. Brahms, Variations on a theme by Haydn for two pianos, op. 56b https://www.youtube.com/watch?v=PeYEcZQvBkU