For the registry, yes. All users need the reg settings. I had IT push the reg settings corporately. There are more settings then just the Portal Name that you can set, and I find it all to be a massive QOL improvement for both me and my users.
Okay, so your SAML connection has "Gas GIS Enterprise - Test". Good to know. Since your login screen has a different name, that means your default of "Gas GIS Enterprise - Test" is being overridden. This is the same behaviour you see when specifying a Portal Name in the registry settings (i.e., it would override the Portal Name specified in your screen cap).
That leads me to believe that you are overriding the Portal Name somewhere. I would at least check the Registry Settings in case it somehow got set there. But if it isn't there, it's presumably been set in Portal Admin... but honestly, I'm not sure where it would be set there if that's the case.
EDIT: On the off chance this is helpful, I found: https://community.esri.com/t5/arcgis-enterprise-portal-questions/portal-login-returns-machine-name/m...