Dear Ronnie,
You mention the process from category to infinity-category. Actually that was why we introduced the term infinity-category
This is why I am thinking you could embrace the way the term is used these days: because it follows precisely your use back then, only removing the restriction of strictness. And algebraicity can be restored. See below...
The problem is that there is no unique choice of such retractions, nor is it clear what might be the relations between iterates of such fillers. These considerations led Keith Dakin to the notion of T-complex for his 1976 thesis; somehow `T-complex' has more recently become `complicial set', but nobody asked me. (Groan! Groan!) So it seems that the notion of quasi category as a weak Kan complex still has not captured something about the basic example; but how to repair that is quite unclear.
This has recently been clarified by Thomas Nikolaus in his work on algebraic Kan complexes (which are essentially simplicial T-complexes!) and algebraic quasi-categories: http://ncatlab.org/nlab/show/model+structure+on+algebraic+fibrant+objects He shows that the model category/quasi-category/(oo,1)-category (check preferred term) of all Kan complexes is equivalent to that of all Kan complexes with all horn fillers chosen. And analogously: that the model category/quasi-category/(oo,1)-category (check preferred term) of all quasi-categories is equivalent to that of all quasi-categories with all inner horn fillers chosen. This says that while a Kan complex or quasi-category is not directly an algebraic model for an oo-groupoid or (oo,1)-category, respectively, you can immediately turn it into an algebraic model by making choices, and up to equivalence, the resulting algebraic model does not depend on these choices. Best, Urs [For admin and other information see: http://www.mta.ca/~cat-dist/ ]