Automator or Apple Script to compress 3 folders?

Question

I have 3 folders I want to compress to an archive and put that archive in a specific location. I'm running in to a few different issues depending on how I tackle the problem. All 3 files live in ~/Library/Logs/, one of the folders is the child of another folder within logs. Here's what the current structure looks like:

~/Library/Logs/
    -Phone
    -Cort
    -Media
        -Cort
            -Logs

I want to compress "Phone", "Cort", and "Logs" (Media/Cort/Logs folder) from the ~/Library/Logs/ folder. Ideally I'd like to just have the zip archive laid out like this:

MyLogs.zip
    -Phone
    -Cort
    -Logs

The first issue is that I do not want to use the direct path (macintosh hd/users/MyAccount/Library/Logs/), I'd like to use ~/Library/Log/ since this will be used for multiple different users.

The second issue is that I do not want to have to choose or select the items via finder since the path will never change for these folders.

The ditto command seems to somewhat do what I want, it doesn't change the folder structure, but I can only do 1 folder at a time.

I've tried to use some similar automator setups such as this one Create a zip file using an Automator Service but I can't seem to make it work without the folder selection prompt, no matter how I pass my paths to it. Every other bash script or automator service seems to be zipping each folder individually, which I also do not want.