It seems as though some people are accustomed to saving from the victory screen and are forgetting about the save button that's always in the upper right. I lost a design that way myself.
Don't worry about the My Designs feature for now. When you save, the game tells you the URL of the saved design with the design ID. Your work gets saved, the only thing missing is the connection to the interface that organizes and displays your designs. The design names aren't appearing in the lower right when you go back to the URLs later either - I take this as a sign that the designs probably won't ever show up in the My Designs folder.
Here's a helpful technique: Keep your PM feature open in another window while contraptioning. When you save a design, just copy the URL and PM it to yourself. As you save more designs, use the quote button in the PM to reply to your own messages and add new URLs to the list. You'll always have all of your recent designs in one place, with convenient clickable links. You can even add a description or name so you know what the URL points to.
If the My Designs feature is missing the recent designs when it's up and running again, you'll be able to easily go back to your PM list, open each saved design, and then save it to My Designs.