A transaction coordinator is hired by the agent or the brokerage to manage the file from contract to close. They track deadlines, collect and route disclosures, chase the documents each party owes, and keep the title company, lender, and both agents talking. Their client of record is the agent or the broker, which is the thing you read and it's accurate.
Who pays is a separate question from who the client is. Three arrangements are common. The agent pays the coordinator out of their own commission and you never see it. The brokerage employs a coordinator and absorbs the cost. Or the fee is passed to the buyer or seller as a disclosed charge, which is what your agent is describing. Whether that third one is permitted, and how it has to be disclosed, depends on your state and on the brokerage's own policy, and in a financed purchase the lender also has a say in what lands on the settlement statement. Ask your agent to point you to where in your signed representation agreement the fee is authorized.
$450 is inside the ordinary range for a third-party coordinator. It is not a fee the title company charges, and you'll pay the title or escrow company separately for their work.
One practical note: if the fee is on your closing statement, the coordinator is still working for the agent, and their instructions come from her. Paying the charge does not make you their client or give you a say in how the file is run. That mismatch is usually what bothers people more than the number does.