Does attempting to masturbate and having a drop exit the penis during a Ramadan daytime – with the intention of pleasure but without the exit of semen – count as masturbation that necessitates breaking the fast and expiation?

Masturbation is the deliberate expulsion of semen by any means. What you did is considered masturbation. If what was discharged was semen (maniy), then you only owe a qada' (making up the fast) without kaffara (expiation). However, if it was prostatic fluid (madhiy), then the emission of madhiy during the daytime in Ramadan does not invalidate the fast according to the preponderant opinion. You should repent to Allah and stay away from this sin.