First, you will see I have moved your post to the Photostory area of the forums.
To answer your question: No, you do not need to download and install the old version before upgrading. Assuming you registered the old version when you installed it you will already have a Magix account, which will show you previously purchased that version and will include its serial key. It is the record of your ownership of that version in your account that allows you to Upgrade rather than purchase the software at full price.
When you attempt to buy the Upgrade your account will be checked (just ensure you are using the same email address as is on your account) and you will receive an email confirmation of purchase, along with the necessary serial key for the new version.
However, before purchase, have you downloaded and installed the Trial version of PS 2023 in order to ensure it runs correctly on your computer? Although that version is deliberately limited in function, i.e., you will not be able to access all its features until you buy the upgrade, it will clearly indicate to you whether your computer is able to run the software successfully.
Any problems, feel free to post back. If you do, please ensure you tell us about your computer (please read this post for the sort of detail we really need) along with the full text of any error message you receive.