V1 is defined as the pilot's initiation of the first action (e.g., applying brakes, reducing thrust, deploying speed brakes)
Is "initiating" maximum braking the same as actually achieving maximum braking?
You're playing with words. Initiating is not the same as achieving, and should be obvious - look up a dictionary if in doubt.
What happens if you "initiate" maximum braking "by" V1 but you nonetheless exceed V1? Do you have another decision to make?
No. Continue the reject. The aircraft accelerating past V1 with the thrust from the good engine/s prior to them achieving idle/reverse is taken into account. Note that reverse though is not on its own taken into account when the aircrafts reject ability is certified, it's just a "bonus" item.