Terminology regarding cross-immunoreactivity
Everything you state is correct, except that I disagree with your statement that a "[...] virus variant itself is a protein, which is composed of peptides.

A virus is typically composed of a protein capsule with genetic material on the inside. So technically, a virus is not a protein, but its outer shell is composed of proteins. Further, a protein does not necessarily have to be composed of (multiple) peptides - it can be a single amino acid chain.