I wouldn't say "I got cold".
I would guess that, "I got cold" is just poor English.
"Cold" can mean temperature or a disease.
(the medical term is influenza, but when you become too cold or wet, you catch influenza, so English people say "I got a cold" and actually to be very thorough, influenza (flu) is a worse disease than a normal little cold.)
I want to put my coat on because I am (I have become - I was - etc) COLD
I have caught A cold because I didn't wear my coat.
I hope that helps.