Shouldn't I be receiving messages on my iPhone?

My phone is off right now due to some miscommunication at my job. I thought that I can get messages as long as I'm using wi-fi. I thought that if I receive texts out of range of wi-fi I would be able to get them when I get in range of wi-fi.

Friends and family have told me they send me messages but I'm not getting the ones I miss from not being connected to wi-fi. I thought that that was what was supposed to happen?

No, you can only send and receive iMessages over Wi-Fi

That only works with Imessage