Here’s how to use "term" correctly in these school-related contexts: 1. Term just started → This is grammatically incorrect because "term" usually needs an article.
Correct: The term just started. 2. It was the start of term → This is common in British English, where "term" is often used without an article in certain expressions.
Correct in British English: It was the start of term.
Correct in American English: It was the start of the term. 3. The beginning of new term → This is incorrect because "term" needs an article.
Correct: The beginning of a new term.

(1) THE term has just started. (2) It was the start of THE term. (If you put the season in front of it, you can omit the definite article, ie, It was the start of spring term.) (3) The beginning of a new term.

All are fine EXCEPT the beginning of new term. It needs the article. A new term, the new term. It was the start of term is general, you could be referring to any term. Start of the new term highlights the fact you’re referring to a specific one.