SMS/MMS is a system service, and all SMS apps are just interfaces to it. So whatever SMS app you use you will see the same SMS/MMS messages. There are many different apps: traditionally manufacturers have installed their own (though it's becoming more common for them to just install Google's one), there are a vast number of third party apps, and some apps that use different protocols can also be used for SMS (e.g. Signal). But they all use the same SMS/MMS database, so it's purely a matter of preference which you use.
(RCS "chat" messages are different: only Google Messages and Samsung Messages support that, because as well as pushing that protocol Google have not provided public access to the API. Why they want to promote this but also limit access it is incomprehensible to me, but that's Google for you).
So it sounds like you have just used whatever the default app was on whatever phone you've been using.
I've never seen that symbol, but I guess it's something to do with the Fold. As I don't own one of those, and don't use Samsung's app anyway, I have no idea what that means. Is there anything different about the messages that show it, e.g. being MMS rather than SMS?