I can use my Barclay's Card Reader (with one of my Barclay's cards, of course) OR insert my Nationwide card in the same (Barclay's) machine and it works too. How is that seeded?
The reader itself is dumb. Its just a user friendly interface to the processor on your card. Hence the reason you can use it with any card that meets the requirements....
The chip on your card is actually a microprocessor. It can contain multiple applications, one of which can be the Mastercard Chip Authentication Program or Visa Dynamic Passcode Authentication. Each application has its own set of cryptographic keys.
So when your card is inserted into the terminal, it makes sure the authentication program is available on the card.
You then enter your pin, the device sends it to the card for processing, which returns the cryptographically derived result using the private key held on the card. This is displayed on the screen on your device.
Your bank has your public key which it can use to validate the output.
The underlying method is probably along the lines of the incrementing counter method outlined above (i.e. the chip on your card holds the current counter value, and the bank has a record of what it reckons the current counter value should be).