It is very simple. when the user / group / channel / bot created in Telegram, the telegram associates with it the identifier of the fixed number (for example, -1001003054104).
This id is persistent, but the user / group / channel / bot attribute (e.g. name , avatar , ...) can be changed. Also, the type of channel (public / private) can be switched the way you want at any time. Switching the type of channel from private to public or from open to private does not affect the participants or previous messages of the channel.
So, you can easily switch your channel to a personal one, create an invitation link, copy the join link, and then open the channel again.
I wrote visual help on this subject on this page.
Another method: If you are using the desktop version of the telegram, you do not need to convert the channel type! Just go to channel info > edit > edit public link > private channel to see the link to the connection, then just click the link!

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