I'm born in Pusan. Nice to meet you. So, I know the dialect 경상도 사투리.
YES, i think, most of 경상도 사람들 will say, 같이 than 함께.
같이 will be sounded [가치] by 경상도사람's tongue.
well, I can agree that 함께 is sounded more polite, 같이 is sounded fondly.
And, THEY ARE COMPLETELY THE SAME MEANING.
And, all the way, you said, can be used.
ALL ARE RIGHT.
there is only one different is sounds.
hope it may be help to you..