They're not the same thing, and the difference matters for how the money moves.
The first document is the buyer representation agreement between you and your broker. That's the one required in writing before touring since August 17, 2024, and it states what you owe your broker. That obligation is yours regardless of what the seller does.
The second is whatever gets the seller to fund it. Two common routes. One is a seller concession written into the purchase contract or an addendum, where the seller agrees to credit a dollar amount or percentage toward your costs, and you then apply that to your broker's fee. The other is a separate compensation agreement signed between the listing broker and the buyer's broker, which sits outside the purchase contract entirely. Which form your market uses depends on the state association's forms and local practice, so ask your agent to show you the actual form they'll use.
The third document is the settlement statement at closing. Whatever route was used, the disbursement appears there with the brokerages named, and that's what the title or escrow company actually acts on. You're right that nobody wires on a handshake.
One wrinkle worth raising with your lender early rather than late. If the fee is structured as a seller concession, it may count against the limits your loan program puts on how much a seller can contribute to your side of the table. Those limits vary by loan type and the treatment of buyer-agent compensation has been actively debated, so confirm the current handling in writing with your lender before you set the number in the offer.