Yup, British-educated here: programme for sequences of activities, program for the thing the computer runs... it’s just a little oddity I’ve noticed I’ve picked up and seem to be quite consistently applying.
Yes... and I use ”to programme” to denote the activity of planning activities, and ”to program” to indicate the process of coding instructions into a computer. I would probably use ”programme to program” if I ever had to discuss the idea of planning one’s intentions to issue instructions to a machine.