Placebo Paradoxes

When you research medicines, alternative medicines, even medical treatments that do not involve a physical substance, you will see reference to the "placebo effect". Statements like:

X is no better than a placebo.
Y is probably just a "placebo effect".
Treatment X is just a placebo.
Medicine X failed to beat a placebo in controlled clinical studies.

What is a placebo? What is a placebo effect? Are they real, fake, or something else? Is the placebo effect magic?  No, but placebo effects are a paradox. Merriam Webster defines placebo effect thus:

"improvement in the condition of a patient that occurs in response to treatment but cannot be considered due to the specific treatment used"

Placebo effect is a real effect in response to a treatment that cannot cause the effect.

A paradox.