I do not think so, but it's confusing. The OIF Triumph tanks I see here all have the rim, but Gary's does not seem to have it. Of the OIF BSA tanks here some have the rim, others (like mine) do not have it.
As the small bolt holes for the Triumph logo are wider apart than those of the BSA-logo, I presume that when these holes are correct, the tank is correct for the specific bike, whether it has a rim or not.
The Triumph has the taps more forward than the BSA, so I would say my tank, without the rim is definately a BSA-tank.
You are correct about the week spot in the tunnel. Mine is also welded there, and the weld is covered with epoxy resin. These tanks need the brace between left and right, in the front bottom.

Ger B