Activity recording starts at my house even though unit is turned off.

Possible that the device is not off, but rather "asleep" in extended tracking mode. If so, you should see the led blinking periodically (I don't remember what color LED). 

It is also possible that this is just an artifact of a partial activity on the device. The device will connect the end point of the partial activity to the point where you resume the activity. I always clear the activity right after I save the recording. 

My first guess is that you have Auto Start set on the default of on. If you turn on device at home it starts recording, and then if you turn off without discarding the recording, when you turn back on at the trail it continues from where you turned off at home.

Go to Setup Menu, then Recording, then turn Auto Start off. You will have to manually start recording when you get to trailhead, unless you are also using Tracking. Starting Tracking will start recording.

It is always best practice with any Garmin GPSr to CLEAR the current track log at the location you want the new recording to start, for example, at the trailhead when you start your hike, and then SAVE the track log at the end of the hike, not when you get home.