drop 3rd strike

[smbbcoach99]

Bottom of 7th. Babe Ruth tourney game. Score 2-2. Team b (batting team) with runner on 3rd 2 outs Batter ha count of 1-2. On pitch, runner takes off for plate. Batter swings, misses, catcher drops 3rd strike. Runner crosses plate ahead of tag, batter steps back away from plate to allow runner in. Catcher then (a) tags batter for 3rd out OR (b) batter then takes off for first and catcher throws to 1b for 3rd out.

Does run count in either situation?

Thinking i read somewhere that the drop 3rd strike is on te same interpretation as a ground out from shortstop to first. But not sure

Thanks