You are correct. Standby(reserve) is not considered rest and for the company to imply that it does is wrong.

When ever you are on-call you're not on rest, nor are you on Duty. Duty is any activity involving flight. A good way to look at it is, if you can't drink a beer then it isn't rest. The pilot must be able to count back 24 hours and be able to find at least 9 consecutive hours of rest, 8 if reduced.

I am 135 scheduled and this is how our standby days work. Not sure if it's the same for non-scheduled.