What is the name for closure-type functions that use the call-site environment?
You are talking about static vs. dynamic scoping of variables. Most langauges today have static scoping, but famously the early LISP had dynamic scoping.

Scoping is a general feature of a programming language and is not something that a closure does (although I suppose one could organize things so that it does). In any case, look up "dynamic" and "static" scoping.