I get the interpretations on this play (3-1 move) can be wide ranging.
Obviously a balk under "Official" rules
NFHS 2-28-5 DEFINED...A feint is a movement which simulates the start of a pitch or a throw to a base and which is used in an attempt to deceive a runner. By the NFHS definition a feint is literally an arm action. Once you have the definition you would apply it evenly wherever the word is used in my opinion.
This was sent to me from a case play...On the 3-1 move, the pitcher has fully satisfied the feint to third as long as he has stepped towards third and feinted a throw with his arm. As such, he does NOT need to disengage while reversing course and throwing to first base. However, it is important to realize that the pitcher is still engaged with the rubber so the pitcher must throw to first in this case or be subject to a balk since the pitcher is not allowed to feint to first base. If the pitcher does disengage while feinting to third, then the pitcher may feint to first because his status is now that of an infielder.
And also...
SITUATION 8: With runners on first base and third base, the pitcher legally steps toward third and feints a throw. With his pivot foot now off the pitching plate, the pitcher turns and feints a throw to first base. RULING: This is a legal pick-off play by the pitcher. (6-2-4a,b)
SITUATION 9: With runners on first base and third base, the pitcher legally steps toward third and feints a throw. With his pivot foot still engaged with the pitching plate, the pitcher turns and feints a throw to first base. RULING: This is a balk. While engaged with the pitcher’s plate, the pitcher may not feint toward first base. The runner from first is awarded second base and the runner from third is awarded home. (6-2-4a)
To summarize:
During this move, if the pitcher disengages during the feint to 3rd base, he may then legally feint or throw to 1st base. It is not a balk.
I would add that obviously the pitcher must separate ball from glove immediately during any disengagement which would be likely following such a move.
My interpretation is that to satisfy a feint under NFHS definition, arm action is required. Its commonly called here.