You asked What is the rule.
It should be clear from the answers and comments so far that there isn't a single rule: it depends on the circumstances, the intended meaning, on the manner in which the answerer chooses to express it, and possibly on whether they are using British English, American English, or some other variety.
First issue: Collective or individual agreement
You question whether when a group gives its word, it is one collective word that is given rather than each individual's? That would depend on the circumstances. If the group were all together at the time and collectively agreed, it would have been a collective or shared agreement, or collective promise. But if all members of the group were asked individually and all agreed separately, then you have one promise from each person.
Second issue: Plural forms
In this usage, word is not a literal word consisting of individual letters of the alphabet, but rather is a metaphor for promise. Promise has a plural of promises, but word in this abstract meaning does not have a separate plural.
On the other hand, you could not really use promise for collective agreement, because (in this context) promise would be understood as somewhat stronger in meaning than word, and could really only be given individually, not collectively.
So now we can have:
Collective (or shared) agreement: They gave their word (singular for the reasons given above)
They each gave their word (singular, because each is singular)
They each gave their promise (ditto)
They each promised
They all gave their word (singular, because there isn't a plural form when used with this meaning)
They all gave their promises (but, personally, I don't think that is a likely expression to be used; and also it is ambiguous because it is not clear whether they each gave one promise or multiple promises)
They all promised (the briefest and most likely expression)
They gave their lives
As Edwin said in a comment "Give one's life is idiomatic" - and that is the standard expression. Idioms do not always follow standard 'rules' and may not strictly be grammatical, and the answer may just be "Because" (i.e. that's just the way it is). Nevertheless, here I can argue that it is grammatical:
Although they may or may not all die together and simultaneously, they each had one life to give, and all of them gave their individual lives. So:
They (all) gave their lives (as for promises above).
First, it's not necessarily appropriate to compare this with the expressions discussed above because they are idioms and this isn't.
Secondly, the Wikipedia reference you mention does not give a definitive answer, and there do appear to be differences of opinion - so what makes you think you can get a definitive answer here?
Brad has already mentioned that abstract nouns do not always take a plural form. With abstract nouns that rarely require or 'need' a plural form, different users may adopt different answers because it is unusual enough for there not to be an accepted 'standard'.
For a variety of different reasons as discussed, you cannot always project from one particular usage to another because:
idiomatic use may not follow rules;
you need to consider collective v. individual;
there may be no common accepted standard;
there may be difference usages in different parts of the English-speaking world; and
that's the way English is: one of the rules is that it doesn't follow the rules and is not always logical!