While I am pretty good at history and believe that Constantine converted to the Christan faith, I can not state that as fact. It certainly looks like he did and he is on record as saying the believed in the one G*d, the Christan God. But it is also noted that he made such statement when it served his political advantage.
Re meteor, no one knows what Constantine and his army saw that day, only that it was an unmistakable sign from G*d and caused Constantine to fight under the banner of the Christan God. Something a Roman elite would find unthinkable at that time. A clever ploy by Constantine? A true conversion? Beats me, but something happened, something big. The body of evidence seems to point at conversion. Maybe papers written by Constantine himself will be discovered and we'll find out.
I heard he continued to pray, etc., at his old pagan altar, but that the next emperor and forever after was Christian. He needed soldiers and there were a lot of Christians by then, so he let them in.
I seem to remember having read somewhere that Constantine did convert to Christianity, but it was at the very end of his life....practically on his death bed.