Religion aside, when and why did people start identifying themselves as Christians? Instead of Baptist, Catholic, Presbyterians, etc. ew
The actual date of the use of the term "Christian" to define the followers of Christ or, more essentially, to differentiate them from Jews, is the subject of an intense and unresolved historical debate. All agree, however, that the term was in use by 150 CE.
"Catholic" or Catholicus means "universal" and "orthodox", the Latin version of the Greek oikoumenikos (οἰκουμενικός) which comes into English as "ecumenical". Claims of orthodoxy were particularly important to the Roman see, and they were particularly keen to reject claims of the Patriarch of Constantinople that he was an "Ecumenical Patriarch". However, at first they did not claim the title "universal" for themselves. Gregory I (d.605) explicitly rejects it. Half a century later, however, Gregory VII (d.1085) adopts the title, and Catholicus becomes very frequent indeed.
The other terms you mention are products of the Protestant Reformation and only come into being in the 16th century or later.