Here's a VERY simplistic explanation of the Campbellite Churches, which they all prefer to call the Restoration Movement, by the way. There are three main church groups which all more or less owe their beginnings to Alexander Campbell. On one hand you have the Churches of Christ (the non instrumental group) and on the other is the Christian Church (Disciples of Christ) and then you have the Christian Churches AND Churches of Christ which are somewhere in between. The Disciples Churches are usually a bit more formal and have a denominational structure which is entirely missing in the other two. Each of their congregations is fully autonomous, and answers to no real central organization. As to whether the third group calls itself a Christian Church or a Church of Christ is entirely up to the congregation.
So, seeing a sign over the door which says Church of Christ can mean that it is a non instrumental congregation, or perhaps an independent Christian Church by another name. One way you might be able to tell is if that sign includes the message, The Church of Christ Meets Here. That is a clear indicator that it is a non instrumental church.
Seeing the sign Christian Church over the door can mean that it is a Disciples church, or it could be an independent Church of Christ by another name. Am I making myself clear?
Christian Church (Disciples of Christ)
Church of Christ
Christian Churches & Churches of Christ
I think This site tries to make some sense of the whole thing.
Naturally, the Church of Christ and the Christian Churches & Churches of Christ sites are unofficial, given the nature of their church organization. There may be better sites, but those looked promising.
Then there is the Church of Christ, Scientist and The United Church of Church which are not really connected to the others at all, but I won't get into that.