A repo transaction and a reverse repo transaction are opposite sides of the same transaction.
A capital market participant enters into a "repo" with a counterparty, who is in a "reverse repo". A repo is repurchase agreement. The counterparty (liquidity taker) is effectively borrowing funds from the capital market participant (liquidity provider) on a secured basis. They sell the security to the capital market participant and simultaneously agrees to repurchase the same security. The difference between the price sold (lower) and the agree upon price to repurchase the same security represents the interest paid on the secured loan.
Higher interest rates discourage borrowing (lower interest rates encourage borrowing) in that higher interest rates increase the cost of borrowing money.
While the Fed and Banks are major participants in the repo market, the repo market has many participants (money market funds, mutual funds, institutional investors, etc). The consistency is that the borrower or liquidity taker enters into a "repo", and the lender or liquidity provider enters the same trade as a "reverse repo".
When the Fed is looking to inject liquidity (increase funds or money supply) into the market, they are engaging in a "reverse repo" transaction. When the Fed is looking to take liquidity out of the market (reduce funds or money supply), they are engaging in a "repo" transaction.