What should I expect when Booting my MacBook Air in Safe Mode?

Here is information about booting in safe mode and what to expect: Start up your Mac in safe mode - Apple Support

If that's what the system is reporting after booting Safe Mode with all the indications confirming it, then I surmise it's a bug in macOS. If you are so inclined, report it: Feedback - macOS - Apple. Be sure to include all details including the specific model Mac, etc.

After rebooting normally (not "Safe Mode") then the output will still be kern.safeboot: 0 and the other indications ("Safe Boot" in red at the login screen etc) won't be present.

I would have had the same reaction. Boot Mode should indicate Safe.

Your other observations indicate it was in fact operating in Safe Mode, and I would be inclined to conclude the screen snippet is in error.

At your option, out of curiosity, please post the output of the following Terminal command:

sysctl -a | grep safe

1 = safe boot, but it might still be erroneous.

If whatever problem you wanted to solve has been solved I wouldn't get too concerned about that Safe Boot flag since I suspect it may be a bug in macOS, and most likely a bug that affects only certain hardware. The other indications you report ("Safe Boot" in red at the login screen, overall clunkiness, etc) is enough for me to conclude Safe Mode is in effect despite the output of that command or anything else.

Have you ever seen Safe Mode on that Mac? Even once?

More curiosity: what is the Terminal output of

nvram -p | grep args

(It might not return anything, which is ok)