No, that won't help.
You must have been an early adopter of Apple mail. Therefore, like me, you have an old user@mac.com acct. Because of the way Apple has transitioned those old accts over the years, an old user@mac.com acct can also receive email for user@me.com and user@icloud.com.
However, not everyone using Apple mail has an old user@mac.com acct. Those who joined later got an acct that only receives mail for user@me.com and user@icloud.com. And those who joined later still got an acct that only receives mail for user@icloud.com. None of those more-recently registered accts can receive mail for user@mac.com.
It's the fact that you have an old user@mac.com acct that seems to be causing the problem, not how you are accessing your acct. The funny new headers seem to appear only for users who have one of those old user@mac.com accts. And it doesn't matter which of the three valid forms of AppleID you use to access the msgs. You will still get the weird new headers.
However, if you had one of the newer accts that only receive email for user@me.com or user@icloud.com, and cannot receive email for user@mac.com, then you would not have this problem.
At least, that's how it appears to me. I have both types of accts: an old some_name@mac.com acct and a separate, newer another_name@me.com acct, which cannot receive email for another_name@mac.com. The weird new headers appear only on my old some_name@mac.com acct. They do not appear on my newer another_name@me.com acct. At least, not yet.