The closest term for that is a stickler.
I have never heard the phrase "Jack-in-office" before. There are lots of words that are close to what you explained, but not exactly what you describe.
Following the rules very closely, while missing the point of these rules is called "Following the letter of the law, not the spirit." This is a very common expression. To my knowledge, there is no word for a person who does this.
Another word that is close to what you are explaining is "Stickler""stickler". However, a stickler generally refers to somebody who follows and enforces the rules under any circumstances,under any circumstances, not somebody who's missing the point of the rules. For example, a professor who refuses to accept a test because it was turned in 1 minute late would be called a stickler. Stickler is also slightly derogatory, but not vulgar. It's also informal.
Some related words from this thesaurus page are "perfectionist""perfectionist", "nitpicker""nitpicker", and "disciplinarian""disciplinarian". These are all related, but I think stickler is closer to the word you are looking for.