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You have misread the linked page:
When traveling from any of our international destinations to the United States...
(emphasis added)
But you have misunderstood this:
by which they mean at what time people should arrive at the airport for departures to various international destinations
(emphasis added)
Given your misunderstanding, it is natural that you ask
For which step between "arriving at the airport" and "boarding the flight" does a passenger to BogotΓ‘ need longer than a passenger to Bermuda?
But by now, it should be clear to you that the answer is "for any and all of them, because the check-in times are for passengers flying from different airports."
What happens if I arrive at the airport later?
You increase the risk of missing your flight, perhaps because of long lines at the check-in counter, security screening, or passport check.
Might they deny taking my suitcase?
Yes, at some point, they will close check-in. There may be a point where they close check-in for checked luggage but will accept travelers with only cabin bags.
What factors do influence whether arriving later is ok or a bad idea?
There are many factors. Time of day is a big one because lines are generally shorter at certain times depending on the airport's schedule. If you have flown from a given airport before, you are more likely to have a sense of how seriously to take their recommendation. For a new airport, it's probably a good idea, for the sake of caution, to take it very seriously.