Your first sentence is an example of what is sometimes taught to foreign learners as the Second Conditional, which envisages an unreal, unlikely or imagined situation. In these sentences, the verb in the if-clause is in the past tense. The verb phrase in the main clause is usually made up of would followed by the plain form of the main verb, but other modals such as could are also found.
Your second sentence is an example of the First Conditional, which predicts a likely result in the future if the condition is fulfilled. Its typical structure is present tense in the if-clause and will (or won’t) in the main clause, but can is also possible.
The term 'conditional subjunctive' is not normally used.
If they are lying, they'll be punished.
Valid, but it means "If they are lying [as of now], ...". This is different from "If they lie, ...", which can be called the use of the generic present, which is not constrained to the past, present or future, because it can be used when referring to past speech, presently ongoing speech, or future speech. The present continuous "are lying" can only refer to speech up to the present (including ongoing speech).
If I'm meeting them, I'll let you know.
Valid, but it means "If I decide to meet them, ...". I'm not sure why, but it could be because "am meeting" here refers to the point (possibly in the future) at which the decision to meet was made. Strangely, I cannot quite pin down the reason this does not apply to the previous example, but I've native speaker certainty.
If you were meeting them today, you would have gotten to know them better.
Valid, but probably not in the context you want. The "would have gotten to know ..." states the definite outcome under a given possibility in the past, and so the "were meeting" has to denote such a possibility in the past. It cannot simply mean "If you [had] met them today", because then the meeting would be conceptualized as a single event. Instead it would probably mean "If they were the ones you were meeting today, ...".
If I had been talking with him, I would have told him.
Valid, and normal, because "had been talking with him" describes a possible past situation (of having been talking with him), under which it is claimed that "I would have told him". Note that it conveys a different meaning from "If I had talked with him, ..." which simply refers to the past event of talking with him.
Best Answer
The phenomenon mentioned here is often called Conditional Inversion in the linguistic literature. Here's an interesting paper about it.
One difference between Conditional Inversion and if is that inversion is really only possible with those three verbs, as you note, and hence is usually only found with counterfactuals. Another difference is that inversion doesn't work well with the focus adverb only:
Those are my judgements and the judgements of the authors of the paper I linked to, and my guess is that they are in line with modern usage generally, though it would be good to check. The Iatridou & Embick paper gives some more potential contrasts. Their conclusion is that verb-initial conditional clauses can't be focused, and that "The use of inversion is meant to indicate the fact that the truth of the proposition in the antecedent is old [information]".
A side point is that in the recent history of English this type of inversion was possible with a lot more verbs, including could, would, might and did (Denison 1998: 298-300). This might account for the overall rather formal flavour of these examples, as Greg Lee mentioned in his comment.
Still, in a large number of situations the two constructions are completely interchangeable (for me at least).
Ref: Denison, David. 1998. Syntax. In Suzanne Romaine (ed), The Cambridge Grammar of the English Language, vol. 4: 1776-1997, 292-329. Cambridge: Cambridge University Press.