Good question. I think it's just an oddity/ambiguity of English (and other languages as well, I'm sure). I know that calculus (with derivatives and integrals) has been called "the calculus" in the past. I mostly just say "the lambda calculus" because that's what I've heard.
Another way that it seems consistent in my mind is that there are many calculi (systems for computing things) out there, and the lambda calculus is one particular calculus that uses lambda expressions as the building block. The pi calculus is another example. The "the" feels a little more appropriate when "lambda" is seen as a modifier specifying which calculus I'm referring to.
In other cases, I'll drop the "the", though. For example, I might say "There are three types of lambda calculus expressions.".
Your final case isn't specific to "lambda calculus". In that sentence, you're using "lambda calculus" as a modifier for the general term "expressions", rather than referring to it on its own. Consider that you would say "There are three types of ceiling lights", but you would still say "the ceiling".
Another way that it seems consistent in my mind is that there are many calculi (systems for computing things) out there, and the lambda calculus is one particular calculus that uses lambda expressions as the building block. The pi calculus is another example. The "the" feels a little more appropriate when "lambda" is seen as a modifier specifying which calculus I'm referring to.
In other cases, I'll drop the "the", though. For example, I might say "There are three types of lambda calculus expressions.".