You do not need to log out on the old device to transfer the progress to the new device, even cross platform.
The option to reload PVZ2 progress onto the new device has been made easy many years ago by using the email register method.
Below shows the simple steps how this is done.
1. On the old device, start PVZ2, click the cloud button on the main menu to enter the register account menu.
2. Register your account with a valid email address. (Once the register is complete, you will be given a small amount of gems as a reward if this is the first time register.)
3. On the new device, perform the same account register using the same email address as above.
4. After email register, you will be given a set of code (to your mail box) for verify you are the account owner, once this is successful, you will get an Identity Conflict screen and be given a choice either to recover the old progress or start a new game. At this stage, please ensure to select the original ID with your original progress and confirm account recovery.
Please note: IF you do decide to start a new game, your old progress will be over written.