install and uninstall postgres

[MalikRumi]

Output of brew config

HOMEBREW_VERSION: 4.2.11
ORIGIN: https://github.com/Homebrew/brew
HEAD: d7d4c8266210c024b93a450a7d357cec0b46a1bb
Last commit: 13 days ago
Core tap JSON: 11 Mar 05:57 UTC
Core cask tap JSON: 11 Mar 05:57 UTC
HOMEBREW_PREFIX: /opt/homebrew
HOMEBREW_CASK_OPTS: []
HOMEBREW_MAKE_JOBS: 10
Homebrew Ruby: 3.1.4 => /opt/homebrew/Library/Homebrew/vendor/portable-ruby/3.1.4/bin/ruby
CPU: 10-core 64-bit arm_firestorm_icestorm
Clang: 15.0.0 build 1500
Git: 2.39.3 => /Library/Developer/CommandLineTools/usr/bin/git
Curl: 8.4.0 => /usr/bin/curl
macOS: 14.3.1-arm64
CLT: 15.3.0.0.1.1708646388
Xcode: N/A
Rosetta 2: false

Output of brew doctor

brew doctor
Your system is ready to brew.

Description of issue

I am new to both Macs and homebrew. I installed postgresql 16 with brew a few days ago. However, none of the usual defaults I expected (like a user postgres and its database) were there. I was advised to uninstall and then reinstall. In the course of doing that, I navigated to /bin and, sure enough, now all my psql commands worked as expected - but only from bin, despite putting postgresql in the $PATH as instructed.

brew info postgresql
Warning: Formula postgresql was renamed to postgresql@14.

This warning did not make sense to me since, obviously, I had not installed 14 but 16, and surely brew should know that because it did it.

==> postgresql@14: stable 14.11 (bottled)
Object-relational database system
https://www.postgresql.org/
Not installed

More confusion. What does 'Not installed' refer to? pg14? Any pg at all? I can't tell, but I looked and pg 16 was still there, exactly where I had last seen it, and there was no pg14 to be found.

==> Caveats
This formula has created a default database cluster with:
  initdb --locale=C -E UTF-8 /opt/homebrew/var/postgresql@14

Not true. I ran initdb, there is still no pg14, or any database cluster based on it, at that location or anywhere else, and what’s more, how can brew claim there is a default database when it just got through telling me it was not installed???

brew uninstall postgresql
Warning: Formula postgresql was renamed to postgresql@14.
Warning: Formula postgresql was renamed to postgresql@14.
Warning: Formula postgresql was renamed to postgresql@14.
Error: No such keg: /opt/homebrew/Cellar/postgresql

Why three warnings instead of just one? Are these coming from three different locations?
I confirmed once again that all I had was pg16 in the same location - and not uninstalled.
/opt/homebrew/Cellar/postgresql@16 does in fact exist, whether it is a 'keg' or not.

Then I went to your docs to look up 'uninstall', but most of that was aimed at brew developers and not at all helpful to me. So I came here. And now you know everything. How do I uninstall all this - safely, or course, and the correct version?