In the phrase
We’re all used to (some-special-word) to mean something different.
would it also be correct to switch the infinitive with the gerund form, resulting in:
We’re all used to (some-special-word) meaning something different.
I believe you are confusing two different constructions here.
Both employ the verb use in the broad sense “do habitually”. And in both, the verb is used today only with the -D form: used.
In one construction, used represents the past tense and is used with the marked infinitive of another verb—that is, the infinitive form with the particle to in front of it. In this case, used may mean 1) simply “did habitually”, or it may imply 2) “did then, but no longer do so”:
used + [to-infinitive]
1. When I was a kid we used [to swim] in the lake.
2. People used [to believe] that the world is flat.
In the other construction, used is a past participle employed as an adjective. It is ordinarily preceded by a form of the verb be.
ADDED: ... or, as Paola observes, a form of the verb get. But this would be deprecated in formal contexts — a form of become would be preferred.
In this case it is still employed with to—but now to is a preposition and must be followed by 1) a noun or noun phrase, which may be 2) a gerund or gerund phrase. In this case, used bears the sense “accustomed” or “inured”
be used + to + [NP]
1. As a sailor he was used to [hardships].
2a. We are very used to [being misunderstood].
2b. I am used to [Mike borrowing my bike].
That yields this, which I think is what you intend by the revised example in your question:
We are all used to [democracy meaning something different in different contexts].
Note that in both of these constructions, the collocation used to is pronounced with an unvoiced /s/ rather than the /z/ called for in ordinary uses of used. In dialogue you’ll often see this indicated with “useta*. In effect, used to is becoming a distinct modal, like hafta or wanta.
However, as this NGram shows you, this is not Standard English. It may become Standard English someday, but it’s not there yet.
I'd say that the correct sentence is the second one.
In fact, the expressions be used and get used are formed by a verb followed by an adjective and a preposition, normally to, which in turn may be followed by a noun or by a verb. As a rule, when a verb is preceded by a preposition, it is used in the gerund.
So, I would say
I'm used to riding my bike
by which I mean to say that riding my bike is no new thing to me, I've done it often, but I could also say
I'm used to Mike riding my bike
which indicates the fact that Mike frequently takes my bike and rides it, and I don't complain about it (or no longer do it...)
As for the possibility of using the preposition of instead of to in these sentences, I suppose it could be a regional form, but personally I've never seen it used.