4G Icons

Hello to everyone, I am new here. My wife has a Galaxy S3 phone and when we are at home and connected to our wireless network, the icons at the top show nothing about 4G. When she shuts off the wireless network and her phone hooks up with the AT and T network, it shows a 4G connection. My phone which is a HTC 1x plus will always have a 4G icon at the top, whether I am connected to wireless or not, the 4G icon is always there. When I do turn off the network and connected to the ATT and network, it shows a picture of a satellite. Is there a problem with my wifes connection? My opinion is that its not showing anything 4G when she is on our home network is perfectly fine and as long as it shows 4G when she is on the ATT network, all is fine. We have different phones and they just display icons different. Is this nothing to worry about? Thank you for any help.