What is the exact meaning of a Predicate, decidability and computability?
There is an unfortunate clash of terminology here. Before exposing it, let me just mention that decidable and computable are synonyms. You mention several concepts, including:


Partially computable: A partial function which is computed by some program that doesn't necessarily always halt.
Partially decidable (terminology I personally have never heard): A language (i.e. set of words) that is enumerated by some program.


These are different concepts. Partial computability is a property of (partial) functions. Since languages can also be viewed as predicates, partial computability can also apply to languages, though this is somewhat misleading, since a partially computable predicate is always computable.

Partial decidability, usually known by some of the other names you mention, is a property of languages. Partial decidability can also apply to predicates by considering the language of all input tuples satisfying the predicate.

The halting problem itself has many variants, of which you mention two:


The halting predicate $HALT(x,y)$, true if the program encoded by $x$ halts on input $y$.
The halting problem, which is the language of all pairs $(x,y)$ such that the program encoded by $x$ halts on input $y$.


The halting problem is a language whose corresponding predicate is the halting predicate. Both are partially decidable but not partially computable or computable.