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.

peptides (2-50 amino acids) are smaller than proteins (50+ amino acids)... it's impossible to call a complex protein a peptide, it fails to meet the definition.

Is that an standard classification? It so simple I can't believe this is the first time I'm hearing it.

Insulin is a peptide hormone. It is a small protein made of 51 amino acids in two chains linked by disulfide bonds.

Probably better people to explain it. First sentence on wiki ” Insulin (/ˈɪn.sjʊ.lɪn/ ⓘ;[5][6] from Latin insula 'island') is a peptide hormone produced by beta cells of the pancreatic islets encoded in humans by the insulin (INS) gene.”

Seems like it is very much on the edge of categories thus fits both. And it really is not a giant as proteins go and instead is on smaller end.