The “dropped third strike” rule applies anytime that first base is unoccupied and/or there are two outs AND the catcher fails to secure the pitched ball. A wild pitch can allow the batter to reach first as long as it is a strike (the batter can swing at a ball far out of the zone for a strike if we wants too).
A dropped foul ball is scored as an error if, in the official scorekeeper’s opinon, the player reasonably should have been caught it. I’m pretty sure, anyway.
Until about 1950, catchers were charged with errors on dropped third strikes where a batter reached. It was not ruled a wild pitch or passed ball.
Mickey Owen’s famous dropped third strike in Game 4 of the 1941 World Series was immortalized in his NY Times obit as a “passed ball”, but it wasn’t. It was an error.
Pitchers DO get a lot of errors, though, considering the number of plays they make; pitcher fielding percentages are generally quite low, in the range of .950 to .960. Pitchers as a group have the worst fielding percentages of any position except third basemen.
Randy Johnson, of course, is arguably the worst fielding player to ever be a successful major league baseball player; going into 2005 his career fielding percentage was an astoundingly bad .899. No, that is not a typo.