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only some of the reminders that I have on the iPhone show up on my laptop. Most do no not, even thought the appearance of the app on the laptop makes it seem that this is the new reminders app.
MacBook Air 13", macOS 10.14
If you upgraded your iPhone to iOS 13.x and you initially used the Reminders app, you were "warned" that if you upgrade your existing reminders that only another iOS 13.x device or a Mac, running macOS Catalina, will be able to access them. You had to agree to upgrade. It was not done automatically without your consent. If you did agree to upgrade that is why you are not seeing your reminders on a pre-Catalina Mac.
If you upgraded your iPhone to iOS 13.x and you initially used the Reminders app, you were "warned" that if you upgrade your existing reminders that only another iOS 13.x device or a Mac, running macOS Catalina, will be able to access them. You had to agree to upgrade. It was not done automatically without your consent. If you did agree to upgrade that is why you are not seeing your reminders on a pre-Catalina Mac.
macOS Catalina is available. Whether or not, your iMac is compatible would be the next question. To find out, take a look at the following article: Upgrade to macOS Catalina. - Apple Support
So, I willed to upgrade my iMac to Catalina, which does not exist yet?
Concerning OS13 reminders upgrade
