Yes, even after turning the machine on and off.
“Initialize machine” is the homing sequence, which will make sure that the machine starts from a known (and precise) mechanical reference, and the coordinates of the zeroes are set with respect to this home position, so the zeroes won’t be impacted by (re)initializing.
There are a number of factors impacting the re-positioning accuracy after a power cycle/homing, but I can tell you this:
- proximity switches have a pretty good repeatability. YMMV but it’s a number expressed in tens of microns, check out this thread for (lots of) details
- if your machine is mechanically sound, the accuracy of the zero across power cycles is negligible in 95% of the cases. It may be noticeable if you power cycle the machine in between doing something that requires a very high precision, e.g. between the cutting the two sides of a double-sided job in metal or some other material where minute positioning errors are visible.
It’s still a good habit to try and NOT power cycle the machine in between two jobs using the same zero, but if you do, it’ll work.