To me, the word clinic is more similar to the term outpatient facility at the end of your sentence. Note that usage might be very different in the UK and Canada.
You go to a doctor's office to see a specific doctor. You go to a clinic to see whichever one of a group of doctors is available, in order of arrival. This could be because you need after-hours care (night/weekends); or because you do not have an assigned primary care doctor (if you lack medical insurance); or because you need a specialized procedure (a mammogram, perhaps).
Confirming Kelly B's cution about different terminology in different countries. The procedure would not be carried out in a doctor's office in the UK (certainly I wouldn't have any procedure carried out in most of the UK doctors' offices I've been in ).
Doctor's Surgery is not really used in North America. Most people would understand what it meant, but if you are aiming at a specific cultural group it would be better to choose "Surgery" for UK use or "Office" for North America.