Meaning and etymology of "down with"

I've searched a lot and found out that down with as a slang phrase means "being in an agreement with something". On the other hand, I know that it also means "death upon something".

So in a sentence like

Down with war!

how am I supposed to know which one of these meanings is applied?

Has this term changed its meaning with time?

This is a simple application of the UP/DOWN Metaphor frame.

As it says in the link above:

What’s UP?
English speakers (like all humans) are oriented vertically with respect to a gravitational field, so the up/down dimension is significant, and English uses it in a variety of metaphor themes. All of them are coherent, i.e, we tend to think of them in the same ways (e.g, LESS, SAD, WEAK, PASSIVE, and WORSE are all negative evaluations, and vice versa.)

a) UP is MORE (DOWN is LESS):
- The prices are rising/falling.
- The stockmarket’s moving up/crashing.
- Turn the volume up/down.

b) UP is HAPPY (DOWN is SAD):
- He’s depressed.
- feeling really up/down about it
- What a downer!

c) UP is POWERFUL (DOWN is WEAK):
- upper/lower classes - superior/subordinate
- the highest levels of the government
- oppressed masses

d) UP is ACTIVE (DOWN is PASSIVE):
- The computer is up/down.
- Are you up for some handball? - Rise to the occasion.
- Down in the dumps

e) UP is BETTER (DOWN is WORSE):
- higher/lower animals
- He fell down on the midterm.
- a rise/fall in performance
- aim high - upwardly-mobile

f) UP is ABSTRACT (DOWN is CONCRETE):
- He’s got his head in the clouds.
- He’s got his feet on the ground.
- Come back to earth.
- higher mathematics
- high-level cognitive functions
- low-level details
- new heights of abstraction
- down-to-earth solution

We already have a question covering the origin.

As to your question, "how am I supposed to know which one of these meanings is applied?" The bad news is that English doesn't work on "supposed to" (though good writers do), so there are indeed words and phrases whose meanings include some that are diametrically opposed to each other.

The good news is that this isn't one of those cases.

The exclamation "Down with X!" would state an opposition to X. If I say it, then I am down upon X.

The description of someone, often oneself, like "I'm down with X", means I am in agreement with X. Or if X is a group, not only do I like them, but I get on with them generally. Likewise "he's down with X" and so on.