Confusion with Push and Fetch in Low Power Mode for Mail on iPhone

When in “Low Power Mode” Settings tells me that Push and Fetch are Disabled for Mail.

I thought this meant unless I open the Mail app, causing a Manual Fetch.

However, in Low Power, off charger, off WiFi, Mail not opened, I still get badged on new mail ???

What then is disabled ??? Is there a difference between ios Mail and Gmail reactions to LPM ???

[Re-Titled by Moderator]

Original Title: Low Power v. Fetch/Push