I have an iPhone 6 running iOS 12.4.1.
I have a paltry data plan ($$$), so I've turned off iMessage in Settings -> Messages.
iMessages show up in blue in Messages app; regular text messages show up in green.
So far, so good. I can still send/receive messages when I'm out and about, and I'm not leaving data turned on and thus chewing through it quickly (and have it available if I need it, such as for driving directions).
Now, here comes the part I don't understand. Although I'm connected to my home network almost the time, much to my surprise, when I do turn on data, I get a number of queued messages. They all seem to be group messages, and show up in grey in Messages.app. [These groups may have been created before I switched off iMessage, if it makes a difference]
This is unbelievably frustrating. I had no idea I was missing messages. I don't get it -- why don't they come through as normal text messages, or come through over my WiFi? Why are they only available when I turn on ridiculously expensive cellular data?
More to the point, is there a way I can get them to come over WiFi, so I don't miss them and don't pay extra for them?