There is a global catalog mode but individual products with an empty price should be in “catalog mode” as well. You can google this to verify.
The product that I was referring to does not have variations.
I created a new product and set nothing on it but the name and assigned a category (the price is empty by default) and the New badge displays. What did you do in your test to make the New badge disappear so you could test the Free badge?
I’m currently testing this functionality because half of my website will be for sale items and half will be catalog items, and I need to ensure that it will all work. So far it looks like the buy button and price are hidden everywhere they need to be on the catalog items.