[AtlUmpSteve]

Without a current USA manual, there is still the obvious (to me) answer. PU did the correct thing, the play at 3rd (when the throw goes directly to 3rd) belongs to the PU. Far too many umpires stay at home with no play just because a runner is coming home, and think that the play at 3rd isn't their's.

On the hit, PU should move up the line in foul ground 1) to create an angle to the possible catch/trap call, 2) to create a lesser angle to the tagging runner on 3rd, and 3) to the holding zone, reading where the initial throw goes. Reading the throw, if directed to home, PU returns to the plate, if directed to 3rd, PU takes the play at 3rd. PU has catch, tag on 3rd, initial (after the catch) play at 3rd, initial play at home, play at home after an initial play at 3rd. If the runner coming home would be obviously safe, a good/great PU would ignore the throw and take any play at 3rd, even if the throw heads home and is cut or returned to third.

BU has tags at 1st and 2nd, any/all plays at 1st or 2nd, plays at 3rd only if/when the initial throw is to home and a return or cut throw is to 3rd. When PU comes up to 3rd (so you don't have to get there!!, and should be nearer 2nd than 3rd), BU is expected to turn and watch the ball to see the next play developing (how else would you know if it's going to 2nd or 3rd??), and see the timing for the runner scoring (or not!!). On this play as described in the OP, your partner would have his back to the plate, you (BU) should be where you can see ball, third, and the plate. Your only possible play is now at 2nd; you should be inside, near 2nd, to the 1st base side (out of the throwing lane of any return throw); now turn and watch to make that call.