When you refer to your grip,I have heard the thumb on top of the stick being referred to as the french grip,palm down as german and somewhere in between as american,In thunder of drums , John Bonhams bio,they said he played one hand french and one german alot of the time.I have seen many drummers use french grip on the ride cymbal for certain figures.
When I first heard of these 3 distinctions, back in the '70s, it was in connection with timpani playing. The French grip was indeed held with the thumb on the top, so when you are looking straight down at your hand, you see the thumb. But it's more than just a position, it's also how you use the back fingers, a lot more finger control of the stroke than wrist. German was explained to me as pretty much the standard snare drum grip, back of the hand up, more of a wrist stroke than fingers.
Then a graduate student came in to North Texas who had been studying Timpani in Germany for 3 years. Completely destroyed what I had thought German grip was. Yes, the back of the hand is up, but the sticks were held very loosely, back fingers completely off the stick, and the stroke was a Moeller type. The mallets were also very different, very thin shafts with very large, puffy heads. Extremely relaxed, very dark tonal sound.
American would be what I was taught by Kal Cherry of the Dallas Symphony. Closer to the French, but the hand slightly tilted to the inside, so the muscle between the thumb and index finger is on top. The stroke, however was completely different, a wrist rotation similar to what I do now with 4-mallet marimba using Stevens technique.