Which is the correct way to obtain administrator access for the installation of Homebrew?
The normal way to do things is as per your third choice ie use an admin account to install things.

However Homebrew's normal setup does not stick to normal Unix conventions of installing excutables as root or a special user which would allow equal access of the installed code to all users, it is designed so that you do installs as one user and so do not need to use sudo/su for anything. However to do this it needs to make /usr/local writeable by the normal user and you need admin rights to that as in the way you did in the end. I think Homebrew really is designed for a computer which only one person uses and has admin rights.

If you do not want to (or cannot) set your user to admin then there are two ways


As per the instructions you quote "If this sucks for you then you can install Homebrew in your home directory" Then the access to ~/usr/local is always for the user
A version of your 3. The Homebrew script leaves /usr/local and all under it owned by the user doing the installation. With the admin user you can change the owner to your normal user by chown -R usr /usr/local


Yes there is only one /usr/local in the file system. Each path is unique. The way to have the "same" directory for different users is to put the directories under your home directory e.g. ~/usr/local