For a written work yes, in colloquial use no
It is not unusual or problematic in spoken discourse to refer to damage as "points of damage", even if that is not a defined game term (unsurpisingly so, as the Style Guide tells the writers not to use that term).
Health is measured in hit points. It is only natural to think of damage as points, then, too. Indeed, older editions of the game used the term, for example on page 9 of the original 1e PHB, it says
Thus, if a hit would normally score 1-6 points of damage upon the opponent [...]
So there certainly is usage precedent, and I doubt anyone would have difficulty understanding you, if you used the term. It does not sound wrong. No problem.
When it comes to writing, the question is, as in all writing, who your audience is. You are translating only for your players, so it does not really matter much, as long as all of you understand what is meant.
If however you would want to self-publish an adventure or setting book for general consumption, then you should aim to stick as close to the writer's guidance and precendent of the published, official books as possible. This will both make it easier for your audience to understand your work, and it will look more professional.
Lastly, if you are working on a book to be publishded for D&D by WotC, then it would obviously be problematic if you did not care for the written writers guidelines you are supposed to follow. You would cause extra work and cost for the editors. This would not look unprofessional in the end, because someone would have to fix it, but it would be unprofessional.
Why the guidance?
It clearly promotes consistency to give guidance either way so you don't end up saying 10 damage on one page, and 10 points of damage on another. Note that the guidance predates the actual rules books, so the absence of the term is a result of the guidance. If the guidance would have advised to use "points of damage" instead, than all the rules would be written that way now. Leaving out the extra words is shorter, and can help to conserve a bit of valuable real estate in the printed books and in stat blocks, where space is often tight. But we do not know what the design reason for the guidance were.