No, the Activated vs Idle status in Alarm.com has no impact on alarm triggers. Those are statuses in Alarm.com only based on activity which occurs while the system is disarmed. “Alarm” is a separate status the panel determines.
However, a wireless motion detector will have a short onboard reset period between when the last motion occurs and when it can trigger again, a battery saving feature. This is unrelated to the ADC activated and idle statuses, to be clear.
When testing for alarms, you’ll need to leave the room out of view of the motion detector, wait a couple minutes, then re-enter. If the detector is in a spot where you didn’t trigger it on the way to arm the system it you can test it right away.