There are many methods
The introduction of queens is an important part of beekeeping, but so many beekeepers have little or no knowledge of how to do it. I once gave a whole day of beekeeping tuition that was attended by a couple who had been keeping bees for over 20 years. Neither of them had ever introduced a queen and hadn't a clue how to do it. I asked them why and the reply was "we have never had to do it". I then asked them what they would do if they had to introduce a queen and they said "we would look in a book!"
I know the couple were members of a BKA that had a teaching apiary, so I wondered why they weren't taught about a common subject like queen introduction. This raises two obvious issues, the first is the standard of teaching and the second is there are many methods of queen introduction, so which is the beekeeper to choose?
In my experience the introduction of queens is more difficult than it was when I started beekeeping. My view is there may be connections with the queen problems many beekeepers have been suffering since the turn of the 21st Century.
In my early years of beekeeping the common method of introducing a queen was to take a queen out of a colony or split a colony and introduce the replacement immediately. We put a queen and about 3-4 attendants in a cardboard matchbox with the lid slightly ajar, then lay it on the top of the frames with the opening downwards. The bees in the colony would chew a hole in the matchbox and release the queen. This could be done at any time, whatever the weather, time of year or nectar flow and was highly successful. It is now much more difficult
There have been many queen introduction methods devised with whole books written on the subject. Most beekeepers have their favourites, but that doesn't mean they are the best. The best methods are those that suit you and give you most success.
If possible I try to avoid introducing queens because of the uncertainty of success. I try to introduce queen cells if I can, so the queen emerges and mates from the colony she is going to head.
Some methods are known by the names of the originator, or as so often happens in beekeeping, the first person to write about someone else's idea. The buttons at the top left will give you a few different methods, some of which are similar. It would help to read the information on the "Principles of Queen Introduction" page.