Baseball's development had nothing to do with math.
Engineering is not a subfield of math either, it merely uses math as a tool. As a field, engineering evolved in parallel with math, only borrowing mathematical methods when their suitable applications were discovered.
Computer Science is a subfield of math because it was developed by mathematicians as a direct descendent of algebra and the study of algorithms, which date back to the ancient Babylonian and Greek methods for division, computing the GCD of two numbers, finding square roots, etc.
It seems kind of strange to me to draw the lines based on anthropology. If there was alternate universe in which a philospher with no mathematical training invented Turing machines and so forth, would you consider CS a subfield of math in one universe but not the other?
Your classification seems as reasonable as any, but the lines seem fairly arbitrary to me.
Engineering is not a subfield of math either, it merely uses math as a tool. As a field, engineering evolved in parallel with math, only borrowing mathematical methods when their suitable applications were discovered.
Computer Science is a subfield of math because it was developed by mathematicians as a direct descendent of algebra and the study of algorithms, which date back to the ancient Babylonian and Greek methods for division, computing the GCD of two numbers, finding square roots, etc.