Here is the entire definition, in one sentence: a peptide is a short chain of amino acids. That is it. Half the mystery disappears the moment you stop thinking of peptides as a category of drugs and start thinking of them as a structure.
The easiest way to hold it in your head: amino acids are single building blocks. String a few together and you have a short chain — that is a peptide. String hundreds together and you have a long chain — that is a protein.
Everything in the body that does something — enzymes, hormones, structural tissue — is built from these same amino-acid building blocks. A peptide is just the shorter version of that chain.
That is why so many peptides feel "natural": they are not exotic chemicals, they are short chains built from the same blocks your body already uses.
✦ Key takeaway
Peptides = short chains of amino acids. Proteins = long chains of the same blocks. That is the whole subject.