Samsung - How do I get notification sounds to work with vibrate only?

I'm not sure you can with the standard approach. There's an assumption you want to be alerted to your phone calls and notifications the same way. But you should be able to hack around it.

Years ago with a previous phone, I was trying to work around a bug where the phone would vibrate, even on silent. The solution was to create a "silent" sound file. Then if you set the phone to ring only and set the "silent" sound file as your ring tone, the phone magically became silent. I think a similar hack could work for you.

Set the ring mode to Sound, enable vibrate while ringing, make a "silent" sound file your ring tone and leave a "not silent" sound file as your notification sound. This might get you what you're looking for.

If you're not looking to have the phone calls just as vibrate only all the time, there's a few profile type programs out there that could help. I've used FNP Audio Profiles before. Seems to do the trick without demanding excessive permissions.

As for the sound file, I had made a short segment of silence in Audacity and saved it as an MP3. You could probably also find something online. Or shoot me a PM and I'll hunt down my old file for you.

In messages you basically have to set every person that texts you their notification sound to silent, go in the text and click on the 3 dots and click on notification sound and pick silent.
Go to phone>settings>call alerts and ringtones, set the ringtone to silent and vibrate while ringing to on.