Correct. The batter got to 1st and stayed at first.
F5 was playing a ways off 3rd base bag and after fielding the ball they ran toward the runner, F6 covered 3rd base. F5 was just running toward the runner (on contact the runner took a few more steps away from third) as they saw the runner was in 'no mans land' and wanted to prevent the run from scoring. As F5 was running towards the runner, the runner went back in the direction of 3rd. The runner got too far away for F5 to tag them and therefore F5 threw the ball to F6 (who was covering 3rd). F6 went to tag the runner but did not catch the ball clean and thus dropped the ball while attempting the tag. The runner gets back to 3rd safe, but their momentum takes them off the bag. Meanwhile F6 now picks up the ball and tries to tag the runner but the runner instead runs back to 2nd base, because it is open.
I wished we would have had it on video. LOL