18.22 Track Database for Runners

Did you mean this here?

No, that is your own courses. There is supposed to be a world wide database of running tracks (I guess the 400 meter track and field type) in the newest firmware version. I haven't seen it either.

^{support.garmin.com/.../
1} These watches will automatically review the Garmin community track-run database to compare the lane starting and end points. If you show up to a track that has already been registered in our community database, your watch should lock-in right away. If it is not in our database, the watch will calculate the lane after approximately 4 laps. 

That implies need a phone and data connection when at or running the track to look for the data.

Track runs are generally the main time i DONT carry a phone on me.

This is a Track Run app/activity only feature.  There is no public/user interface to the database.  Now you can skip the "calibration" process for each new track that you visit.  I've tested on two of my local tracks without the need to calibrate.  I got perfect tracking snapped to the lane I was running.

No phone needed, just your watch. 

Is there a way to check if any of the local running tracks are included in the database or not?

I don't believe so.