If I update a Contact Name from "test" to "test_123" then the new name should get updated to existing tickets in IM/ SD/PM/Change Module.
Sounds like you are learning SM the way I did. These forums are a great resource, but can be diffacult to navigate. Especially when you start getting into much older versions or even different applications altogether. If you haven't found it, I have found some good tips in the Help Center. You should be able to download it separetly if it wasn't installed with your implementation.
In regards to you specific question: I had an issue where I created new fields for users in Change and realized much later that I did it wrong and needed to go back and update open and closed records. You may be able to use some of that to help with your situation. Just read through the thread because it evolves:
edit: Link to post