I was always taught that when you are talking about the time you fell
asleep, you say "I went to sleep at 10pm". I am increasingly hearing
people say "I slept at 10pm". Is that also correct?
"I slept at 10pm" can mean that you went to sleep at that time; you might say "I went to bed at 8pm and finally slept at 10pm." It means, in standard English, that you were not asleep before 10pm. Another use of slept would be "Last night I slept for eight hours."
Also, "I slept at 10pm" doesn't mean you went to bed then, although it has been used that way among some speakers of English as a second language and among children whose parents are not native speakers, according to the answers in the link you provide. Answers in that link also suggest it's slang common among the Z (post-millennial) generation.
If you are asleep at 10pm when, for example, someone telephones you, you would say "I was sleeping" when you called.
Insomnia is a common enough problem that there are standard ways to express going to bed, going to sleep, falling asleep, having slept, and so forth.