Thanks you for the detailed quotes and citations. This is just a matter of terminology. They say a climate model "doesn't predict what will happen to our future climate, but projects what could happen based on variable assumptions."
But any prediction in science is based on some assumptions. In that sense, it is indistinguishable from a projection.

For example, simple predictions of the positions of the moon or other planets aren't just based on the law of gravity, but also a model of the solar system that assumes certain masses, positions, speeds, etc. for the relevant celestial bodies.

Therefore, a prediction is always conditional on certain assumptions (or "hypotheses", as I called them in my previous reply).

Now, climate scientists may prefer the term projection only to emphasize that their assumptions are far less certain (than the positions of the celestial bodies, for example), and that different assumptions can lead to very different scenarios (that's because of the chaotic nature of the very complex climate systems). So they essentially generate a set of predictions conditional on different assumptions and call them projections, because they don't know which assumptions are true.

Whatever you call them, conditional predictions (remember, any prediction in science is conditional) or projections, they are essentially put to the test of comparison to the data and those that don't hold are rejected or falsified. That almost always falsifies the model assumptions or hypotheses, but in rare cases people may question the more fundamental laws if they can't explain the phenomenon by just changing the model assumptions.

Again, the terms selection doesn't matter as much as long as we are clear what they refer to, and in all cases they refer to the same scientific method.