Hi @ten_todd,
You are correct in your assumption, most sites having a full domain the standard Default match detection which is set to Base domain (although this is a configurable default) will match for most auto-fill scenarios for pages.
For those who are not using a domain on a local network per say, I have seen usually the Host or Starts with option typically will work best, and you can also set this matching on an individual login basis.
You can check out more at
Hope that helps 
P.S.
While the above should allow for better URI matching and thus allow the browser extension to detect the proper login for the page, you can also always Manually auto-fill by simply selecting the login in the extension once you’ve searched for it in your Vault.
This would attempt to auto-fill the login on that page, without the need to copy & paste credentials.