What I usually do is a one-on-one copy of all files from one server to another, and then a simple phpmyadmin export and import on the new server. But then there’s no modx preinstalled/running on the new server yet.
From that Serverfault link you posted, I’m interpreting that you are trying to only move over templates, chunks, tv’s etc, while keeping the content of newwebsite.com intact?
I’d imagine just selectively exporting tables from website.com/modx, and deleting those you want used on newwebsite.com to avoid the duplicates.
If you’re running the same version of MODx on both, I don’t imagine this will do harm. Just gotta be careful in selecting the tables to drop