Acts 11:26 regarding the word Christian? Referring to Acts 11:26: Did the people (unbelievers) of Antioch call the "Believers" Christian, or did the Believers start calling themselves Christian first in Antioch?

Here is what I mean: Acts 11:26 The disciples were first called Christians at Antioch.
Acts 26:28 Then Agrippa said to Paul, “You almost persuade me to become a Christian.”
I have this feeling in the tone of these two verses that the word "Christian" was more of a mockery since it came from unbelievers, like we have "goody-two-shoes", especially the way Agrippa said it to Paul, I could almost hear the rest of Agrippa's audience laughing at this!?
And 1 Peter 4:16 "Yet if anyone suffers as a Christian, let him not be ashamed, but let him glorify God in this matter." sort of supports my argument!?

It seems that this mock-name Christian was getting around, and the disciples were resenting it, so 1 Peter 4:16 tells them not to be ashamed (for they mocked our Lord also) but that they should glorify God in this matter!?
Your thoughts fellow Believers?
Strictly speaking, the reference in Acts is to "disciples" (μαθητὰς) and not "believers" (πιστεύοντες), though the latter word seems to be used elsewhere synonymously in Acts (e.g. 5:14).

The Greek literally says (in different word order) "the disciples were called first Christians" (χρηματισαι τε πρωτον ... τους μαθητας χριστιανους).  Perhaps it may be true that the Christians called themselves Christians, but the Greek text does not supply the reflexive (i.e. "the disciples called themselves Christians").  

It seems that even after the disciples/believers were called "Christians", it was still more common to refer to them as those who were of "the Way" (e.g. 19:9, 19:23, 24:14, 24:22).  This is also found in The Didache (English translation here), where the essentials of a Christian life are described as "The Way of life".  The word "Christian" (Χριστιανός) only appears three times in the entire New Testament (including the passage cited; the other two are Acts 26:28 and 1 Peter 4:16).  It is even rare in the writings of the Apostolic Fathers, where it only appears 5 times (once in Ignatius' Epistle to the Magnesians, once in the Epistle concerning the martyrdom of Polycarp, and 3 times in the Epistle to Diognetus).