Both the amide group and peptide bonds is the same but the term amide bond is used for simple molecules and the term peptide bond is used for polymers where a large chain of polymer is formed due to amide bonds as in poly peptide and also in proteins. But VCAA usually accepts both when they say "name the bond" for peptide bond/amine group
I don't think your definition for peptide bond is 100% correct.
A peptide bond is the bond between two amino acids but is not called that for all long polymers.
If you have a di-carboxy (not sure if that's the right name for 2 COOH on the same monomer) and a di-amino (two amino groups on the same monomer), then when they undergo polymerisation to form a long chain, the linkage is just called amide (I believe)
Interestingly, this is how nylon form of you want to go look at more info on this
So peptide is the name for them in protein where amide is the name for them in any other scenario (I beliece)