ANO reference
Amen to this, Genghis.
What I do is have Hobbs start time, takeoff time and landing time, and Hobbs shutdown time entered in the aircraft's technical log. If I don't trust the students' entries, I refer to the TWR logs and replace the takeoff and landing times with what was recorded there.
What I invoice to the customer is /flight/ time, plus ten minutes (or 0.2 hour if they want decimal).
The three times I need to justify as the aircraft owner are:
1. Elapsed calendar time since an engineering event;
2. Engine hours;
3. Airframe flight hours.
1) is a non issue, and 2) and 3) are both available from the Technical Log.
As far as invoicing to the students: flight time plus 10/12 minutes is entirely reasonable, as any other format invariably encourages them to skimp on preflight or after landing checks.
I'm happy to share my Technical Log format for anyone curious who sends me a PM. I keep it as a Google doc spreadsheet, totally easy.