> In proportional representation you would cast your ballot for e.g. The Democratic Party.

You are confusing "proportional representation" (which is a continuous-valued property of many election systems) with "party list proportional" which is a particular election system for achieving a high value of proportional representation.

Notably, Single Transferrable Vote is a system in which people vote for candidates as they do in FPTP elections (except using preference ballots), but which is designed to acheive proportional representation.