What exactly is a private cloud?

I personally think your lecture notes are incorrect on that point. You could do cloud computing without virtualization, provided that there was some way for the service provider to maintain control over the compute nodes sufficient to allow them to be remotely wiped and rebooted

What exactly is a private cloud?

I personally think your lecture notes are incorrect on that point. You could do cloud computing without virtualization, provided that there was some way for the service provider to maintain control over the compute nodes sufficient to allow them to be remotely wiped and rebooted.

(And, yes it can be done easily. Many modern x86 / x86_64 systems have support for booting from a network. All you'd need extra is something to perform a hardware RESET remotely, and that is pretty easy to implement.)

However, as a general rule cloud computing is implemented using virtualization, both with public and private clouds. This is because virtualization allows you to get greater efficiency by share the physical hardware between virtual servers that are (typically) idle most of the time.

Finally, note that there isn't a single true definition of "cloud computing" or of "private cloud". The meanings of these terms are evolving ... just like the rest of the English language is evolving. Asking us which of the various opinions of what (for example) "private cloud" means is correct is ... asking for yet another opinion. There is no authority to decide whose opinion is correct. You just need to make up your own mind what makes most sense to you, and be prepared to change your mind.

If you are asking this in the context of revising for a test or exam, it is advisable to give maximum credence to your teacher or lecturer's opinion. That is just common sense!