[Tanniker Smith]

When they die (even if they didn't show this) the words spoken over the body are "His watch has ended."

You have to assume that over that day he was dead, someone mentioned it.

[Pearls Before Swine]

I agree with your interpretation.

Besides, there’s the old non GOT saying, “Once bitten, twice shy.” “Once killed by the Watch, twice...”

But, I wouldn’t be surprised if Jon is called Oathbreaker by some in subsequent episodes. After all, how many characters are going to believe he was really dead, until brought back by the rites of the God of Light?