When is she out on a failed tag up

[DaveASA/FED]

Ok 1 out R1 on 2nd, R2 on 1st ball hit to right fielder, runners off with the pitch, ball caught by right fielder, R1 (F1 used to CR but coach used all subs that inning to get playing time) WAY off 2nd toward 3rd starts back to 2nd but fails to retouch, about 3 feet from it, as R2 is close to 2nd (she actually tagged up correctly) R1 goes to 3rd, they throw the ball into the infield and to 3rd where they tag R1 standing on 3rd, they then throw it to 2nd and touch the base she did not properly tag from. I ring the out.

I got to thinking should I have rung her when they touched her on 3rd, as I knew they were appealing her not tagging up, or was I correct in making them tag the base to get the out? All this was done live ball, I think they could have taken the ball to the pitcher and then appealed it verbally as a dead ball appeal, is that correct too?