A word for the "urge" to point out others' minor mistakes, not to define this phenomenon

Solution 1:

If you want to go with one word, it might be censorious: (or censoriousness for the tendency/urge)

having or showing a tendency to criticize someone or something severely : very critical

The details are more related to psychology but this urge might come from chronic criticism (or having a critical mind).

Psychologytoday mentions that these people are usually self-critical also and this transforms into chronic criticism of others.

Critical people tend to be highly self-critical. As hard as they might be on others, they are usually harder on themselves. They were often criticized as children, at least implicitly - the message was clear that, in important ways, they weren't quite good enough. Self-critical patterns tend to form in early childhood and by late adolescence mutate into chronic criticism of others.

Also mentioned is the characteristics of a chronic critic:

Feels compelled to share an (unfavorable) opinion on just about everything. Has never seen an artistic masterpiece or a preschool graduation that couldn't use a bit of tweaking. Prone to eye rolling, smirking, and/or slow-motion head shaking, as well as a tone best described as oozing with condescension.

Solution 2:

I would say that someone who has the urge to nitpick but consistently suppresses that urge is a closet nitpicker, nitpicking only privately, mentally.