Andre points out:
"To call a monoidal category 1-braided is kind of confusing because there is no commutation structure on a general monoidal category. A monoidal category is 0-braided." Being an outsider, with no previous neither usage or opinion on this terminology beyond just monoidal and/or tensor category, this seems to me definitive, and more than enough to settle the question.
Well, I agree with Andre's argument but it does not convince me to use Andre's terminology nor John's terminology (see my objections below). The shift of numbers in Andre's terminmology is annoying when you try to prove stabilisation hypothesis using higher braided operads. I hope to talk about this proof in Genoa in a couple of months but it follows readily from another atabilization theorem for n-braided operads. It is where I was more or less forced to call braided operads 2-braided operads despite violation of ("foo" = "1-foo"). Another argument in favor of this terminology is that it provides a uniform terminology in higher dimensions which agrees with E_n-algebra point of view developed by Lurie and also his proof of stabilization hypothesis (see Urs's message). I agree that it creates some clash in low dimensions but I think it is not a big deal since classical terminology does not have numbers (nobody calls a monoidal category 0-braided or symmeteic monoidal category 2-braided monoidal). The low dimensional cases are important but they are not always good models for higher dimension. As an example, -2 and -1 categories as Baez and Dolan pointed out can be understood as one pointed set and two pointed set correspondingly. Should we shift the numbers and call category a 3-category?
But I think "braided = doubly monoidal" is even better. After all, a monoidal category has one tensor product; a braided monoidal category has two compatible tensor products, and a symmetric monoidal category has three.
The trouble is that n-monoidal categories already exist. They were introduced my Balteanu, Fioderowicz, Shwantzl and Vogt. This is why I also see n-tuply monoidal as confusing. I do not say that they sound identical but certainly very close to each other.
But I will not lose sleep if Andre uses "k-braided" as a synonym for "(k+1)-tuply monoidal".
I am glad to join John. I am also grateful to everybody participating in this discussion. Terminology is a very important issue but I do not think it is a crime to use a different one if the clarity of exposition dictates it and if one acknowledges the existence of an alternative. I think I will continue to use my own terminology but I am going to give more explanation in the introduction for those who like a different one. with best regards, Michael. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]