Like many English words, the word "cite" has multiple definitions.
To quote from the American Heritage Dictionary:
1.
a. To quote or refer to (a book or author, for example) as an authority or example in making an argument.
b. Law To refer to (a previous court decision or other legal precedent), as when arguing a case.
2. To mention or bring forward as support, illustration, or proof: cited several instances of insubordinate behavior.
3.
a. To commend officially for meritorious action in military service.
b. To honor formally.
4. To issue a notice of violation to: was cited by the police for jaywalking.
That is, to "cite" can mean to quote or refer to some source. Like, "I cited this book in my research paper." That's definitions 1 & 2.
Definition 4 is very different: To identify a violation of law. We will typically say that the police or a court "cited" someone for breaking such-and-such law.
Yes, "cite" could have been substituted for "write up" in the quote you give. In that cases it's not really a violation of "law" but of school rules, but same or similar ideas.
No, it's not at all the same as definitions 1 & 2. We're not quoting anything or referring to anything. We're formally noticing the fact that someone has broken the rules. Presumably we write this down somewhere, hence "write up".