Infinite Language vs. finite language
Oh my. This seems like a confusion caused by the (old school) terminology of "finite-state language" as a synonym for what is known today as "regular language".

Anyways, the standard definitions for finite/infinite accepted these days regard only the size of the language:


a finite language is any set $L$ of strings, of finite cardinality, $|L|&lt;\infty$. 
an infinite language is any set $L$ of strings, of infinite ($\aleph_0$) cardinality $|L|=\infty$.


A finite $L$ is always regular. 

An infinite $L$ can be regular (sometimes called "finite-state"), decidable (sometimes called "recursive"), non-regular (non-finite-state), non-decidable, etc.,