Time Machine backup with two volumes

Hello everyone.

1. I had two partitions or volumes on my MacBook SSD and I did a TM backup. Both volumes were backed up as expected. Afterwards I decided I no longer needed the second volume, so I deleted it from my MacBook SSD and did another backup. The latest Time Machine backup still includes the deleted volume. I wasn't expecting that. I thought it would now only contain the now single volume on my MacBook?
2. I now have a new MacBook and I would like to transfer my data using the Time Machine backup mentioned above. However, I don't want the second volume contained in the backup copied over. What is the best way to approach this? Can I simply select the appropriate folder for the volume I don't want on the TM backup drive and delete it, or will that cause problems?

Thanks.