How does transaction signing work?

Let me start with what I know. I know that each holder of crypto has a public and private key. I know that information that has been encrypted with a public key can be decoded by a private key. I know that it is one way and is computationally extremely difficult to deduce a private key from a public key.

Now lets say Alice wants to give Bob 1 eth and shes willing to pay the transaction fee to get it included in the current block. She knows her public and private key, and she knows Bob's public key. How does she prove to the miner that she really is Alice and not just some person with Alice's public key? It's not like she can give the miner her private key.

Does the miner use Alice's public key to make a message and have her decode it with her private key to prove she is Alice?

edit: or is it that Alice takes her private key and encrypts the transaction, which can be decoded by anyone with her public key?