Is there a short expression for a teacher that teaches to his/her pupils in their homes, as opposed to in a classroom ?
My guess would be home teacher but I'm not sure (it is unclear if home refers to the teacher's home or his pupils)
In the UK, the most common term would be "private tutor". They might also be called a "home tutor" or possibly "personal tutor", but the teaching could be in any setting, not just a personal home. Being taught individually by a professional but not in a school setting is nearly always referred to as "private tuition".
'Tutor' tends to be preferred over 'teacher' when the training is personal, tailored to the individual.