How to disable Text Message Forwarding between iPhone and iPad?

Thanks. I'm now trying to turn iMessage back on on that iPad. Remember: one of the things I tried was turning Settings > Messages > iMessage off. In order to try what you suggest, I have to turn it back on. I have supplied my Apple ID and password twice, and both times I've received the message that the iMessage server cannot be contacted. One would have thought disabling Text Message Forwarding with the iPad would be as straight forward as enabling it.

Sorry to take so long getting back to you on this. Signing out of iCloud completely on the iPad caused the iPad entry in Text Message Forwarding on the iPhone to disappear. That's what I wanted in the first place, since I'm preparing to sell that iPad.

Now I have a related problem with the Text Message Forwarding on the same iPhone. It shows two devices with the same name, "RRL-MacBook Pro". A long time ago I had only one Apple laptop, a 17" one, so I named it that (without a "17"). And when I sold it and got another, it received the same name. At some point I needed something smaller and got a black MacBook, which became "RRL-MacBook". When I replaced it with a 13" MacBook Pro, I named the replacement "RRL-MacBookPro13". So now in the Text Message Forwarding list I see "Richard's iPad Pro", "RRL-MacBookPro13" and two "RRL-MacBook Pro" entries. Only the entry for the iPad Pro is on, yet on both Macs I can send and receive messages using Messages. Now I'm wondering, (a) how is that possible if I haven't enable text message forwarding for them on the iPhone, and (b) why two "RRL-MacBook Pro" entries.

I suspect iMessage is like an IMAP mail account in the iCloud, and there's a difference between sending a text message via SMS and sending it via iMessage. iMessage is for messaging among Apple ID's. If you don't have an Apple ID or aren't signed in, you can't receive of send iMessages, only SMS. Since all my devices are signed in, they're all able to iMessage. Is this how you understand iMessage vs. SMS?

So what's text message forwarding? Is for allowing a device that is running Messages and signed in with the same Apple ID as the iPhone to send and receive SMS's through the iPhone?

And what about the two RRL-MacBook Pro's? Might it be that I sold the previous 17" MacBook Pro without first deactivating iCloud. I don't think so. I usually erase the boot disk and install the most recent version of OS X possible. I suppose I could either sign out of iCloud on the only laptop with than name that I still own and see whether both entries disappear or just one, or assign a new name. At any rate, I wish I knew which course of action is less likely to "muck up" things. Do you have any thoughts on this?