Urs, Andre did not say that Lurie's usage of the term (infty,1)-category distinguishes from yours but that his usage of the term infinity-category (note the difference) is specific to mean quasicategory: his introduction says: "We begin with what we feel is the most intuitive approach to the subject, based on topological categories. This approach is easy to understand, but difficult to work with when one wishes to perform even simple categorical constructions. As a remedy, we will introduce the more suitable formalism of ∞-categories (called weak Kan complexes in [10] and quasi-categories in [43]), which provides a more convenient setting for adaptations of sophisticated category-theoretic ideas. Our goal in §1.1.1 is to introduce both approaches and to explain why they are equivalent to one another." [For admin and other information see: http://www.mta.ca/~cat-dist/ ]