Yes, that distinction definitely applies to me. Usually when I say "X" it means "I believe X with almost certainty" and when I say "I believe X" indicates that there is some doubt still, maybe a 90% confidence, but not a 99% confidence.

But in that specific case, as Misha said, I didn't need to actually believe it - it was a belief in belief in my chain of thoughts, an attempt to rationalize the initial mistake, that appeared, with further analysis, to not be the real cause of it. Having this as a real belief or not wouldn't change the reasoning.