No, I don't realize that there is a "huge difference" between "being an employee of a company and being contracted by one".

I understand where you are coming from.  You are not the only ones to have heard these very specific usages for "employee", and this is a very common mistake for people to make.  I have worked as a full time employee for American corporations and have also worked as a contractor.  I know how these terms are generally used.  I also know, due to being a native English speaker, that the way words are used in various professions or even individual companies is not the only way they are used and in some cases can even be considered incorrect usage.  In other words, it helps to sometimes assign a more exact meaning to a word AS USED in a profession, or within a legal document, etc, but that does not change nor limit the overall meaning of the word.  

Here, this should help.  It is an example of ONE definition of the English word "employee" (first one that showed up on google):

em·ploy·ee  

/emˈploi-ē/

Noun

A person employed for wages or salary, esp. a nonexecutive.

Synonyms

worker - servant - employe - clerk