The term “First Wizarding War” is never actually used in the books.
The term “First Wizarding War” is never used anywhere in the seven Harry Potter books - searching each of them for the term produces no results. The most likely origin of the term is a continuation from Chapter 38 of “Harry Potter and the Order of the Phoenix” which is titled “The Second War Begins”, in which the Ministry recognizes Voldemort has returned.
“HE WHO MUST NOT BE NAMED RETURNS
‘In a brief statement on Friday night, Minister for Magic Cornelius Fudge confirmed that He Who Must Not Be Named has returned to this country and is once more active.”
- Harry Potter and the Order of the Phoenix, Chapter 38 (The Second War Begins)
The term “First Wizarding War” is likely to originate with readers extrapolating from that chapter title - if Voldemort’s return is called the “Second War” in the chapter title, then his previous rise to power could theoretically be considered the “First War”. However, the first time when Voldemort rose to power is not referred to in this way by either the narration or characters, including in the book with the chapter titled “The Second War Begins”.
“Well, firstly, he wants to build up his army again,’ said Sirius. ‘In the old days he had huge numbers at his command: witches and wizards he’d bullied or bewitched into following him, his faithful Death Eaters, a great variety of Dark creatures.”
- Harry Potter and the Order of the Phoenix, Chapter 5 (The Order of the Phoenix)
Therefore, the reason that Grindelwald’s rise to power was not called the First Wizarding War is twofold - in-universe they are not called the First Wizarding War because nothing mentioned is called the First Wizarding War, and the out-of-universe explanation is that readers extrapolated the name from a chapter title referencing the second war with Voldemort.