Ban Phe is without doubt the least touristy of the three. I spent some time in Ban Phe recently and although there are some Foreign tourists it's mainly a Thai holiday retreat.
It's very popular with Thai school camping trips. Not the prettiest place I've been in Thailand but as near to the real Thailand as you'll get on a TEFL course.
Chiang Mai is very popular with foreign tourists and you have to travel outside the city for sightseeing etc which can be expensive.
Phuket is the biggest tourist destination in Thailand and expensive.
In Ban Phe you'll pay Thai prices.