Dear Andre,
I agree that the terminology (infinity,1)-terminology can be useful.
Okay.
Can I point out that Lurie is calling a quasi-category an infinity-category?
Okay, let's look at Lurie's use of terminology then. Notice that just a little later in On the classification of TFTs http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0465v1.pdf#page=31 In the remark 2.1.26 he speaks of "the various models of the theory of (oo,1)-categories" referring to Julie Bergner's article which shows that quasi-categories, sSet-categories, Segal categories and complete Segal spaces give four equivalent such models. Then still a bit later in (oo,2)-Categories and the Goodwillie calculus http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf he uses terminology exactly as I have been suggesting in my previous messages: starting in the third sentence: "Let us use the term (oo,n)-category to indicate a higher category in which all k-morphisms are assumed to be invertible for k> n. [...] The theory of (oo,1)-categories is also quite well understood, though in this case there is a variety of possible approaches. [...] These are known as quasicategories in the literature; we will follow the terminology of [HTT] and refer to them simply as oo-categories." So, for what it's worth, Lurie adopts the convention that I was talking about, it seems to me: to say (oo,n)-category for the general concept and use other terms for concrete models. He just happens to have the extra convention that "oo-category" (without the ",1") is his term for the model that you called quasi-category. Maybe in this context it is noteworthy that in this last article alone, there is presented literally a dozen of different and equivalent models for (oo,2)-categories. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]