First you have to have the same major version of Lightroom Classic. Since you have Lightroom Classic 10 on one computer, you must have Lightroom 10 on the second computer. But since you have OS 10.14.6, you should install LrC 10 on the second computer.
You should not allow your Lightroom Classic versions to upgrade, this will prevent the two systems from talking to one another.
You place the catalog file on an external HD and move it from computer to computer, so you always have the most current catalog available to both computers.