Powering several amplifiers with one power supply

If you ask 10 EEs this, you will probably get close to 10 answers. This is not a well-defined convention, but is roughly outlined as you guessed. A power amplifier will be able to provide more power than the previous stage. A current amplifier will be able to provide more current than the previous stage.

If you need a definitive answer, I would say ask the person that you are involved with on this project. The key is always to understand what the other person means, not necessarily what they say. Basic communication theory. :-)

For your LED strip you need to know if it is supposed to be constant current or constant voltage supplied. Luckily it says DC12V on the strip itself, so it's constanst voltage with 12V.

You don't need limiting resistors as they are built on the strip (see those three little black things near the LEDs? Those are the limiting resistors).

You power supply has it's specs on the label: 12V 1,25A. So it has a maximum power of 15W.

The voltage is okay. As for the power, that's a bit difficult without knowing the exact specs of the strip, but if we take those you linked, you have calculated correctly - 50 LEDs will take 12W, which is 1A on 12V.

You are supplying the right voltage and you don't draw to much current, so everything is within specification and I don't see how there should be any premature aging happening here if the strip stays cool (glue it on an aluminium profile for example).