Since the "Church of Christ" is the only authorized name for the church, why do some take pride in calling it the "conservative church" or the "non-institutional church"? I also hear of the "mainstream church." Is this not liberalism and denominationalism?
First off, "church of Christ" is not the only name found in the Scriptures. The Bible uses numerous descriptive names for the church, such as "churches of Christ" (Romans 16:16), "church of God" (I Corinthians 1:2), "body of Christ (Ephesians 4:12), "kingdom of God" (Acts 8:12), and many others.
Since the name is not trademarked, there are a large number of groups calling themselves churches of Christ who do not strictly follow Christ. Thus the problem arises in how to talk about sets of churches, who use the same name, yet teach differently. In English, we do this with descriptive adjectives. I might talk about the Honda cars on the road, but if I want to narrow the field I can mention the yellow Hondas or the four-door Hondas.
"Conservative" and "liberal" are just how a person approaches authority. A liberal finds justification in an authoritative document based on what the document does not say. A liberal approach states that when something is not expressly forbidden, then it is allowed. A conservative approach is an opposite. Something that is not expressly allowed, directly or implied, is forbidden. One of the distinguishing factors between churches is their approach to the authority of the Scriptures. Some take a liberal approach and others a conservative approach.
One of the problems with a liberal approach to the Scriptures is that it is hard to establish boundaries of how far you can go and not exceed the authority of Christ. Thus in recent years, there has been a rise of ultra-liberal groups. Most of the congregations have a liberal approach to the scriptures, but not extreme, so they began calling themselves "mainstream" to distinguish themselves from the "way out there" liberals. Most mainstream churches consider themselves conservative when compared to the ultra-liberals, so they might call themselves "conservative" and the ultra-liberals, "liberal." They tend to call the original group that they branched off from "antis" because their truly conservative approach to the Scriptures makes it appear to the liberal groups that they are against everything they want to do.
Because of the liberal approach, most (but not all) of these congregations found justification to use the Lord's money to fund institutions to do portions of the church's work. The conservative congregations objected. Thus, another distinguishing attribute is the support of institutions (institutional) or the rejection of institutions (non-institutional).
The disunity is wrong, but it isn't denominationalism, at least not yet. However, I should mention that the ultra-liberal groups are fast approaching a denominational viewpoint. Denominations tend to see themselves as an organization composed of churches. The Bible speaks of the church as being composed of Christians, whether we are addressing the church in a universal sense or a local sense. Denominations tend toward organizing beyond the local church, so there are regional and sometimes worldwide headquarters. Denominations use documents beyond the Bible to define their particular sets of beliefs, such as statements of faith, creeds, and manuals.
For more on the history of the church, see "What is this business about "anti," "liberal," and "sound?""