I picked the one that was immersed in water.
 I liked the one that was painted yellow.
Stricly speaking, they are both ambiguous between adjectival and verbal passives.
If they are intended to convey a stative meaning rather than a dynamic one, i.e. the state of being immersed/painted was the result of some prior event, then I would say that "...
These are predicate adjectives since the linking verb to be ("was) precedes the phrases and specifies the noun.
The trick to identifying predicate adjectives is spotting linking
verbs. The linking verbs include the following:
....The verb to be (in its various forms, e.g., am, are, is, was, were,
will be, has been, have been).
First, I should point out that the first sentence is an abbreviated form of "testing is complete". This kind of abbreviated form is used in newspaper headlinese, notices and computer status displays.
As an adjective, complete means whole, so the following sentence means that all of the pieces of testing have been put together... probably not what you meant.
Your prolix question asks about when to use the past perfect tense. That tense happens to use a past participle, but there are other uses for the past participle too.
Aside from that, whether you use the perfect tense depends on whether what you are saying needs to be fixed in the past relative to another past event that is part of the meaning of what you ...