When it comes to SSMS, it is exactly what the error message says. You need to be member of db_owner. When it comes to SQL Server itself, it's perfectly sufficient to be member of db_ddladmin, but the crippled Table Designer in SSMS is a very old piece of junk, and it only checks for db_owner. (I can see this by runnin a Trace against SSMS.)
I'm less certainly about Visual Studio. When I test with a user that is only member of db_ddladmin, I don't even come as far as you seem to do, but I see this:
I also tried Azure Data Studio, but this did not work any better. I got the message: An error occurred while initializing the table designer: Failed to initialize the table model.
Thankfully, there is a workaround for all of this: Simply type the CREATE TABLE statements and skip the designers.