John, the push button appeared with the first ejector offered by Baker. I have seen only one example of the first ejector and have forgotten the patentee, but it is on Paragon 874, made in 1905. It is visibly different from the subsequent ejector designs. I think the Baker Quarterly of the 1905 period announced the ejector availability. The number 883 seems somewhat familiar to me, so if I had to guess, I would guess that Paragon 883 pictured above had the second ejector [most common] design. But 883 is close enough to 874 that it , also, may have the first ejector design.
Here are a couple of pictures of the somewhat later ejector with the push rod. This is N Grade 958. I may have seen a couple more of the push rods, but they are not common.