As long as you have that fan-shaped icon, you are on WiFi. Yes, your carrier will appear, but it doesn't matter - the WiFi is running. If/when you lose WiFi, that fan shape will be replaced by a cellular-type icon, such as "3G," "4G" or "E" (among others). The only time you will see "iPad" is when you turn off cellular. I think having the carrier name and signal there is to always let you know what your cellular strength is, in case you need to switch to it.
In all your attached pictures - you are on WiFi.
Hope this reassures.
Marilyn