Well in some cases (Like with Isaac II Angelos killing Hagiochristophorites) It could be that the records available to historians do reference a given date on a calendar for a specific period. The death of an individual, particularly a well known one, would have been recorded or remembered by someone and thus could be pinpointed.

As for the first example, when you have a tradition stated to having been spread, it is probably marked as a specific date to the day that it was first recorded in a text or dated to evidence in some form of record. Finding a writing by say, Herodotus for example that says "X event happened in my 26th year, sometime in the winter, 3 years after this person took power, the third day of the week" Gives historians a starting point. Cross referencing other writings or artifacts from that period of time (Paintings, etc) can give us a better idea of the exact date or time of year... this process continuing to resolve a finer and finer picture until we can reasonably state that said event happened on X day"

Not everything CAN necessarily be pinpointed to the exact day, but a lot of things can just by putting a lot of little things together in a timeline. If a ruler's birth is recorded to the exact date, then another writing mentions that 5 days after this something happened, then 40 days after that event something else happened, and somewhere inbetween... it just kind of goes on and on.

How do we know it happened on that day?

Well it's pretty easy for your second example, actually!

"The hot and hasty Stephen Hagiochristophorites, who cared little for his lord and emperor in various ways, agreed to arrest Isaakios Angelos and after confining him to prison to subject him to that death which Andronikos was to sanction. He arrived at the house of Isaakios near the Monastery of Peribleptos in the late afternoon of the eleventh day of September in the year 6794 [1185]." Annals of Niketas Choniates.

Choniates was a contemporary historian and somewhat involved in the events. And keep in mind these sorts of "this day in history" lists are completely devoid of context. Mr. Hagiochristophorites was notorious in his own time, an important government minister and enforcer of an unpopular ruler (Andronikos). And his death was the beginning of an eventful coup d'etat. If he didn't die in the course of a major event that Choniates saw fit to record in detail, we may never have known his fate.

You're right about the first one, though. The way it's written on the September 11th page makes it sound like it pinpoints the very date of a Christianity-wide cultural shift, which is silly. But if you look at the article it's referencing, it's clearer that the day was the first instance of a practice that later became widespread. See the 5th paragraph here.