SSDT on VS2013 for SQL Server 2012
That is correct. Forward compatibility is guaranteed. Backwards compatibility does not exist*. Having 4 different BIDS/SSDT instances on a machine is a pain I've known all too well but there's not much you can do.

There are ways of constructing SSIS packages such that your solution can target whatever version of SQL Server will be hosting them but that's installing, purchasing and learning a separate product (Mist).