I don't use Time Machine for my backups, preferring to use SuperDuper!. So, I am unsure if there is a way you can have Time Machine do two separate backups to two separate devices.
However, if you use it to make a backup to the partition on your internal drive, then via the link I provided above, you should be able to "copy" that backup to the external drive. However, you proposed setup is risky for the following reasons:
1. Having a backup on your internal drive assumes nothing goes wrong with the internal drive, nor accessing it. That is not always a wise choice.
2. Given that your external drive is 10 years old, there is the distinct possibility that it could go bad, and in fact become completely useless. I'm actually surprised it lasted you that long. What is the brand of the drive?
If I were you, I'd purchase a new external drive. 1 TB external drives are not very expensive. In the US, one can get them for about $40 to $50 US.