A lien is a claim that can affect the property or the money due at closing. The current payoff comes from the creditor or taxing office, not just an older statement. The title company checks title and brings the confirmed payoff information into the closing figures.
When you sell to us, we pay the closing costs. Your liens are paid off from the sale at closing.
Closing costs are separate from the mortgage balance, liens, back taxes, and your share of current property taxes. Those amounts are confirmed for the property and shown in the closing figures; the closing-cost promise does not pay them.