Is this a balk with runners on first and second?

Runners on first and second
Right Handed Pitcher goes into set position and pauses (as required)
Pitcher attempts to pick off runner at second by “spinning to his left, past first base, and directly to second base”. His pivot foot (right foot), remained in contact with the pitching rubber until he turned toward second base.

There’s a a rule that states a pitcher cannot feint a throw to first base.

Did he break the rule here?

As I am interpreting this, he did not break the rule about faking to first. He made his move toward second base (stepped and threw toward second) and not toward first base. The move needs to leave no doubt that his intention was second base. If he’s half-hearted about it or gives up on the move midway between first and second, he’s begging for a negative interpretation by the umpire.

The pitcher may rotate clockwise or counter-clockwise while executing his pick or fake to second base-- it does not matter.

In the situation you cite above, the pitcher may throw to second or fake to second, but he definitely can’t fake to first while remaining in contact with the rubber. Again, the pitcher needs to make it clear to which base he intends to pick or fake.

Another thing to consider is the umpire’s position. I am assuming a two umpire system used in all HS and American Legion levels and below. Some umpires set up in B (on the infield grass on the first base side of second base) and other set up in C (on the infield grass on the 3rd base side of second). If he is in B position, it will be easier for him to determine second base as your intent. If he is in C position (which some umpires prefer when there is a chance of a steal of 3rd base) then he may call a balk if he’s not sure that the pitcher’s intent was truly second base. I’m not saying don’t attempt this if the umpire is in C. I’m just saying you need to make a definitive move if you intend to simply fake the runner back to second.