ORD - 2019 R2 - Is there a way to rename existing points in a Template

No, you have to do it the hard way.  Fastest way is to rename them in the active template portion if the tool.  That was they are all in a list and you do not have to go hunting for them on screen.

The *.itl file that contains all the templates is basically a large text file that allows editing using a text editor. You could try finding and editing the points there.
If they have a specific name maybe you can even try a find and replace.

The ITL is an XML file, and it is possible to 'Find and Replace' - but you would have to ensure that you do not 'break' the XML. The point is defined, it is also used as the 'parent' of other points, and as a point in a Component. 

There is a lot of scope for error in this approach - using Marc B's 'active template' method would be safer, and probably quicker.

I would suggest more descriptive names. When you have to transition between template or assign point controls, the last thing you want is point 6 in one template to really be point 27 in another. I worked on a huge project and it seemed that ever template drop had a bunch of unorganized point names and we had to be very specific with our point control station limits and points numbers. to prevent one control form encroaching on the next template inadvertently. It was a nightmare. And it was difficult to retrofit better names as doing so often broke some point control. However, if one was successful, the number of point controls needed dropped significantly and it was certainly more manageable.