Call it a clarification, then. When I heard that line in "The Vulcan Hello" for the first time, IIRC I immediately wondered "But what about Donatu V?", and I think I saw others ask the same question over the following week. So the reference to Donatu in episode 2 helped clarify matters.
And no I don't think someone on the set was suddenly going crazy over one line in the script and saying, " we have to add this one line to fix that..."
Not on the set, no, since the decision would've been made well before filming began. I don't know about today, but I recall reading that back when TNG was made, the typical process time from the first story outline to the finished episode was about 11 weeks, with the actual filming happening about midway through that span, and with a new one being shot every 8-9 days. So naturally you had a lot of episodes being worked on in parallel at the same time, and of course they'd try to stockpile as many scripts as they could before they actually started shooting, so they'd have a headstart to cope with the inevitable delays. In the case of a 15-episode streaming season, it's possible they had the whole season written before they began filming, or at least the first 9 episodes before the midseason break.
Keep in mind that there was a lot of creative staff shakeup in early DSC, with Bryan Fuller being let go fairly early, so that episode 1 was scripted by Fuller & Akiva Goldsman while episode 2 was scripted by his replacement showrunners, Gretchen J. Berg & Aaron Harberts, with Fuller only getting story credit. So yes, it's entirely possible that the writers working on episode 2 decided that a Donatu V reference was needed to clarify any misconceptions arising from that ambiguous line in episode 1. It's not "going crazy," it's just the normal process of catching problems and fixing them that's a routine part of how writing works.