Can't Download Logic X on new computer

I have 2 Laptops, one old one new.

I've owned Logic Pro X since 2015 on my old computer. I'm trying to download it on my new computer from the App Store but it's asking me to pay for it, there is no cloud icon to download it like there was for other apps that I own.

My old laptop is running OS High Sierra, and in the App Store on that, it gives me the opportunity to update my copy of Logic Pro x to the latest version..

My new laptop is running OS Mojave and in the App Store it's asking me to pay $199.99

Both app stores are logged in with the same Apple ID.

I've gone on "Iforgot.apple.com" to try and check whether Logic was associated with either of my older apple ID's however, both of the ID's return an "x@xx.com is not an Apple ID" error.

I don't want to have to buy Logic again, and it's running fine on the old computer, but I'm retiring that computer and want to be able to download it on my new computer.

Anyone else have this problem?

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