The terminology is all a bit loose. Proteins are peptides, but we don't usually call them that. But when a hormone is a protein, rather than a small molecule, then we call it a "peptide hormone". I suppose because we're making a chemical point, that the hormone is in the family of peptides rather than steroids or eicosanoids or whatever?
As you say, insulin is a fully fledged protein, albeit a small one, and i wouldn't call it a peptide myself. But then, insulin is 51 residues, whereas tirzepatide (Mounjaro) is 39; is that such a big difference? FWIW i think a lot of other "peptides" are even smaller, eg CJC-1295 is about 30 residues.
As you say, insulin is a fully fledged protein, albeit a small one, and i wouldn't call it a peptide myself. But then, insulin is 51 residues, whereas tirzepatide (Mounjaro) is 39; is that such a big difference? FWIW i think a lot of other "peptides" are even smaller, eg CJC-1295 is about 30 residues.