Is this a journey into Smittyland?

[LakeErieUmp]

This one is for my personal development.
I was watching (in the stands, quitely) my son's 13 yr old travel team. R1 and R3. R1 takes off on the pitch. Cather throws to the pitcher for a cut-off play (why pitcher and not SS cutting in I'll never know). Pitcher throws to third to catch R3 who saw the cut-off play and headed back. Ball goes into dead ball territory. By this time R1 has already reached 2nd and has rounded just in case. BU and PU confab and give R3 home (correct) and R1 third.
Wouldn't the throw from catcher to pitcher (now an infielder, not that it matters) be the FIRST throw in the infield, and the pitcher to third base the SECOND throw, thus 2 bases from the time of THROW, not the time of PITCH?
Or is I thinking R1 gets home just me being too ticky-tacky?
Thanks for your input all!