What's the correct name for this type of bath and shower mixer tap?
Google 'bar mixing valve' or 'thermostatic bar mixing valve'.

The one you picture looks to me like it is not thermostatic nor pressure balancing. It has no 'anti-scald' protection. Anti-scald valves are now required for a shower in nearly every plumbing code as far as I know. 

The bar mixing valve type has the advantage that the entire valve is outside the wall, enabling repair or complete replacement without disturbing the wall.