@Kardamom I grew up with those exact ornaments (both kinds)! We even had a tree topper that was very similar to the non-bird ones you linked, but instead of a depression in the middle, there was an actual opening, and in the opening there was something my family called “angel hair”. I suspect is was actually fiberglass filaments… white stringy stuff.
The oldest I personally own date back to the year I was born, so 43 years old. My parents started a collection of Santa ornaments for me, adding to it every year. So I had a pretty good assortment of my own ornaments the day I moved out of my mother’s house. I’ve done the same for my children (though not Santas, specifically). Now that they’re teens, we always pick them out together, a new one for each child every year.