Singular noun objects of plural subjects

What is the rule for singular noun objects of plural subjects? For example I and google N-grams agree that

They gave their word.

is better than

They gave their words.

To my mind, this is because each has but a single word to give and cannot give more. However, google seems to disagree with me on

They gave their life.

vs

They gave their lives.

As you can see in the linked N-gram, the second is far more common than the first. What's going on here? People (cats excluded) only have one life and it seems passing strange that they would be able, let alone willing, to give more.

If pressed, I would claim that the object should only be plural if each of the subjects has many. For example

They called their friends

If each of the "they" in question has only one of whatever object we are talking about I would call it ungrammatical to have that object in the plural form. Am I wrong?

EDIT:

This question was prompted by a comment on this post on S&F.SE defending the use of consciousnesses as the plural form of consciousness by referring to this page. The word was used in this sentence:

Looking for a story where people download their consciousnesses into bodies

It occurred to me that whether or not consciousnesses is correct, it should not be used in that (admittedly bizarre) context since each person only has one consciousness to download. I would have therefore said

Looking for a story where people download their consciousness into bodies

This to me seems like an equivalent sentence to they gave their word or they gave their life. In all three cases I would use the singular form of the noun since each of the subjects can only give (or download as the case may be) one.

In the sentence They gave [object], the object is singular when the group collectively gives a single thing. In your example, They gave their word, the group collectively makes the same promise. In contrast, They gave their lives says that each member of the group devoted or sacrificed their individual lives – even if they shared a common cause.

This is true for all transitive verbs. For example, if a group of comrades make a conference call (or even a series of calls) to a single person, They called their friend. However, if they each make individual calls to separate people, They called their friends.

EDIT: Also note that we don't always pluralize abstract nouns in English. The more abstract the noun, the more likely we are to use it collectively. That's why plural consciousnesses sounds awkward even when talking about multiple minds: We usually think of minds concretely, consciousness in the abstract.

In many cases, you can use a noun either way. Choosing to pluralize or not helps to emphasize whether you mean it concretely or in the abstract. For example, All presidents swear an oath to uphold the Constitution of the United States. This emphasizes that all presidents make an oath of the same nature, even though they all do it individually. They all give their word.

In contrast, we usually write that they devote their lives to upholding the Constitution to emphasize the individual nature of their contributions. You could write that as singular life instead to emphasize the common, abstract nature of their devotion, but we usually don't.

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.

Options

So now we can have:

Collective (or shared) agreement: They gave their word (singular for the reasons given above)
Individual agreement:
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).

Consciousness(es)

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'.

Summary

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!