Learn English – Tense agreement: If I have a chance… I would

Traditionally, it is considered neither, though it is sometimes called a mood for lack of a better word. The word function would seem the best term. I am assuming that you are talking about the word would in your example. The adjective "conditional" just means "related to a condition"; when used as a noun, it usually refers to a function of the past subjunctive tense of modal verbs (would, could, might, and should).

There are three traditional moods in English, indicative, subjunctive, and imperative; there are several tenses, which are combinations of moods, aspects, and temporal properties (you might say past and present are aspects, though they are usually not so called).

There are three types of conditional sentences, as Henry has pointed out, which are mainly just sentences in which an explicit or implicit condition is present. There is a function of the past subjunctive that is called conditional because it is used with one of these types, the so-called hypothetical condition (if he were rich, he would be unhappy). I think that is the one you mean; while it is sometimes called a mood, I find this unclear and confusing: how can something be of two moods, both subjunctive and conditional? Function is the term that both fits best with established terminology and best describes what kind of phenomenon it is.

In your A sentences we have a straightforward situation. Backshifting after a past-tense reporting verb is nearly always correct, even when the statement is a universal truth, so A2 is correct.

Non-backshifting, if the reported statement is still true or (as in first conditional sentences) possible at the time of reporting is also correct, so A1 is correct, as is B1.

B2 is inconsistent. "He just told me that it's important and to give it to you when the time was right." The speaker has started with a non-backshifted 's (for is) and then has gone on with a backshifted was. This would almost certainly go unnoticed in speech. We often start an utterance with no clear idea of how we are going to complete it, and the conflation of two different ideas is common*. In writing, where we have more time to think and correct, B2 would be corrected by a careful writer so that the reported statement would have either two present-tense forms or two past-tense forms.

For example "I have seen him two weeks ago". Here the speaker has conflated the ideas of "I have seen him" and "I saw him two weeks ago and said what no careful native speaker would write.