More or less you got the idea, but not every mixer work the same way. Some of them will give you a private key instead of sending you money from a different address. Giving you a private key is better in theory, but it also carries another risk, because you have to trust the service to not use it asap after they give it to you. In short, they can steal your money but pretend that they don't do it since they already give you the private key, and push the blame on you instead. This is just one aspect to it, I suggest you read https://bitcointalk.org/index.php?topic=5117328.0 if you want to learn more and see which mixing services are prone to attack compared to others.