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?