Other users' coins instead my own?
Yes, you are right. There are no "coins". There is only an ability to write some data (transactions) into one global file (blockchain).

But it is possible sometimes to track who is an author of transaction. For example, I am buying something in online shop (with delivery) and paying bitcoins for it. Now the shop knows my name and my address (for postage). The spending transaction has a "change address". If I use the same address for change - the shop owner would also know (with some probability) the amount of bitcoins in my wallet! Some people may think that this is not good, especially if they have large amount of funds.