When you create a new site using the RAD tools as you have, it is automatically configured to use a connection string called "LocalSqlServer", which is located in machine.config.
To "override" that string, you can just "remove" the reference to the default LocalSqlServer connection string, and create a new one also called LocalSqlServer. That way, you won't have to alter any of the providers to use a new name.
Essentially, it would go like this:
<remove name="LocalSqlServer" />
<add name "LocalSqlServer"