Premier Pro not working between iMac and MacBook Pro. Can't update.

Hello.  I'm here because I need some help and am hoping to find some advise here.

I have a 2013 iMac, which is updated to the latest macOS version (Catalina version 10.15.7).  I have Premier Pro installed on this iMac (version 22.6.4 Build 2).

Recently, I purchased a second-hand 2019 MacBook Pro, which is updated to the latest MacOS version (Ventura 13.4).  I have Premier Pro on this Macbook (version 23.4.0 Build 56).

I have started a Premier Pro project on my MacBook Pro, but wish to finish it on my iMac.  But when I go to open the saved project which I have transfered from my MacBook to my iMac, I get a message saying "This project was saved in a newer version of Adobe Premier Pro and cannot be opened in this version."

I've tried doing a software update on my iMac, but it keeps telling me that my iMac is up-to-date.  I've also tried doing a software update of Premier Pro on my iMac.  It tells me there are two updates available (V23.5) but it's telling me it's not compatible.

So I am faced with two problems.  Firstly, I can't seem to update Premier Pro to its latest verion on my iMac because it says it's not compatible (even though my iMac is telling me its already been updated to the latest version (macOS Catalina version 10.15.7).

And secondly, because of the first problem, I can't transfer a Premier Pro project from my MacBook Pro to my iMac.

Any suggestions solutuons would be greatly appreciated.

Lastly,  if both my iMac and my MacBook Pro have the latest macOS versions, why does my iMac have Catalina 10.15.7 and my MacBook Pro have Ventura 13.4?

Thank you kindly.