The question is all asked in the title: What is a proper/ideal name for the "predicate logic" / "predicate calculus" language. Some people call the language itseft "predicate logic", while some call it "predicate calculus". Which is better, on what aspect? Why?
It depends on how you want to ‘slice your cake’, as it were. Normally, a formal system or ‘logic’ consists of two parts: syntax and semantics. The syntax is further sub-divided into a formal language, and a deductive calculus (i.e. rules (or axioms) that govern the proofs).
Note that the language itself has at least two aspects: Firstly, there are the expressions that are part of the language – the sentences and formulae, the predicates and constants, the quantifiers and variables, the connectives and auxiliary symbols, etc. Secondly, there are the rules that tell you how to construct complex expressions from simpler ones – e.g. ‘If φ and χ are sentences, ‘φ v χ’ is also a sentence.’
Now, which term one uses for which part is mostly a matter of convention. However, I wouldn’t use ‘predicate calculus’ to refer to just the language of the formal system. Rather, I’d use that term to refer to (a version of) the deductive calculus. Perhaps I’d also use the term to mean predicate logic as a whole, but with a focus on the deductive part. If you think that the language itself already constitutes the ‘logic’ – e.g. because it already encodes logical form –, you might say ‘predicate logic’ and mean just the language.
Conceptually, what’s more important to realise is that there is no such thing as the language of predicate logic: each textbook has its own dialect. As far as I know, they all include at least one quantifier, matching variables, connectives, and at least some (n-place) predicate letters. Which connectives they include, already varies from one source to the next. Commonly, individual constants are also included, and brackets might be used. Additionally, you might have function symbols, identity, smaller-than and/or larger-than, etc. etc. The point is that there really isn’t one language that would deserve to be called the language of predicate logic. (This becomes relevant once you turn to meta-logic! Not all meta-theorems apply to all dialects.)
So, the best practice is to give the particular dialect you are working with a name of its own – such as ‘L-prop’. Often, ‘L=’ is used to indicate that the dialect in question includes identity. (In more mathematical contexts, people often start with a particular theory, and then name the language after the theory. E.g., one might use ‘L-PA’ to mean the language in which Peano Arithmetic is written.)
That’s how it is strictly speaking. More casually, it is not uncommon to just speak of ‘the language of predicate logic’, and leave it to the reader/audience/interlocutor to figure out the details.