TLH21
Sometime ago (back in 2004) I was at the airport waiting for a flight to my winter vacation. In the terminal was a grand piano. A young student played a familiar tune: the theme of Mozart Symphony #40 in Gm mvt 1. Sounded like something out of a Faber book arranged for easy piano. I started piano not too long ago but wasn't in the mood to play anything.
The thing about playing at a public piano is that you're not in a recital and don't practice weeks or months ahead of time unless you know there is a piano at a certain location and planned to go there. Otherwise a public piano would be a chance encounter. Just relax and play the piece(s) you know well at the moment. You do expect an audience of some kind but not in a traditional sense where everybody would sit and watch in silence. In an airport or shopping mall many people would just walk by, some would watch for a few minutes while others would continue walking.
Once I was at a second-hand store playing on an old piano. The only piece I could play off my head was a piano arrangement of "What a Wonderful World". Got the sheet music from a piano store a few weeks earlier and played it at home. Do people play from memory? Nowadays it's common to see people with music on a phone or tablet. I find that a phone screen is rather small but people do read off it. The last piece I played at a public piano was a Chopin followed by a Bach Prelude. Not a lot of preparations beforehand.
My version of "Home on the Range" was something that I reproduced off a student performance video more or less by ear. I liked this version with LH arpeggios. There are many easy versions for beginners. The most basic has just the melody line alternating between both hands.
At a piano store I'd try some top of the line pianos like a Baldwin grand or a Kawai K-800 upright selling for around $10,000. After playing a few tunes I'd leave. If I stayed too long I feel an obligation to buy something.
When it comes to recording, it's ok to warm-up by playing under tempo or touch up on certain spots before pressing the button.