Can I add a CVS Pharmacy vaccination record to Apple Health?

The only things that you can add to Health and the Apple Wallet are verifiable information. That is explained in the Related Article that you came to the forum with. Not all information can be added. You may just be able to take a photo of your card and keep it in photos.

From the article you have checked before posting your question:

You can download verifiable vaccination and test result records if your vaccine or test result provider supports this format. Check with your provider to see if they support verifiable health records. 

Source: https://support.apple.com/en-us/HT212752