Unfortunately, no. The way it works right now, person notifications are essentially just a “type” of motion notification. Regardless of whether you have (non-person) motion notifications turned on or not, when the camera records a motion event, it triggers the cooldown period. If that motion event contained a person, the motion notification will say that a person was detected. If that motion event did NOT contain a person, the motion notification will just say motion was detected. If you turn off regular motion notifications and turn ON person notifications, the motion events are still getting recorded, so the cooldown period is still getting triggered, regardless of whether a person was in the motion event or not.
BUT… they just released a new premium feature yesterday called “Complete Motion Capture” that mitigates this to a certain extent. It’s $1.50/month. Using this feature, there’s no longer a cooldown period, although my understanding is that person notifications are still only based on the first 12 seconds of the video. That means that if a person enters the frame on the 13th second, during a “Complete Motion Capture” event, your motion notification won’t mention that a person was detected.
I used it during the beta, but I haven’t signed up for the new service yet, now that it’s officially rolled out, so I’m not sure if any of that changed.
Edit: Oh, I guess it’s already officially un-rolled out? Apparently they delayed it.
¯\_(ツ)_/¯ But anyway, that’s definitely coming soon.