Jim,

Simple answer to your question, from my point of view, NO.

If I order 2" square tube steel (HSS) I am going to get a 'tube' that has a 2" OD measurement on four sides.  I also specify what wall thickness I want.  But, I do not ask for a particular ID.

Pipe is ordered by ID and schedule which is nominal not firm.  Round (HSS) tube is ordered as square or rectangular by OD and wall thickness.

When you look to D1.1, I know you are referencing a different D code, annex K defines 'tubular' as including cylindrical but appears to primarily include square and rectangular.  The welder quals and other sections also give you set ups for both square or cylindrical for testing and fabrication. 

While I am in no way speaking for AWS, because of the continuity they generally apply to the majority of definitions and standards I would have to say that 'tube' includes square, rectangular, and cylindrical. 

But, I also do not have the code you are working to.  I could be totally off base.  Just my two tin pennies worth from your statement that square tube does not have an OD.

Have a Great Day,  Brent