Until there is a way to mirror custom SMS notification sounds to the watch, I have literally been missing responding to texts because e-mails and everything else is "the same beep".
I could not get the suggestions here to work. Here is my work around for now.
I went to the Watch app, My Watch, Notifications, Messages.
Changed from Mirror my iPhone to custom notifications.
Unchecked show alerts.
Yes now my watch doesn't "notify me" when a text message comes in any more. But now my phone always goes off with my per-contact sounds like it is supposed to, and if I want to use my watch to see what message just came in, I just leave Messages docked (My Watch, Dock, Favorites instead of Recents, Messages at the top).
This is not quite as convenient as being able to "raise to see" texts as they come in, but FAR better than missing texts completely, which is what was happening.
This is not a good solution for someone who frequently uses their Apple Watch away from their phone, because now you've just told it not to notify you of Messages.
I pretty much always have my phone, I just wanted the convenience of Messages without digging the phone out of my pocket, plus I already have a well-established set of custom notification sounds for knowing who a Message is from and responding with according urgency (or not) due to the notification sound.