I've been using Photoshop and Illustrator since version 2 - not CS2 but 2.
The recent updates to Photoshop have further destroyed functionality of the Pen Tool. Why does it not work exactly like Illustrator? In illustrator if you want to add a point, just click down on the path between two points, you'll see the (+) and a point is inserted. If you want to remove one, simply click on a point and you'll see the (–) and it will be removed.
Photoshop 22.3.1 is not showing the paths between points if you first create a new path with the [+] in the paths panel like I've been doing for over 25 years. It seems like a bug. In any case, a workaround is to make sure no paths are selected, then create a path. A new paths work path will be created and you can rename and continue drawing on the new path layer as always.
DEV - Please fix the pen tool in Photoshop - make the options EXACTLY the same as illustrator - there's no need for all of these various Pen tool options, and adding extra tools that have no real functionality is not a help - it's a hindrance.