You seem to be conflating several things here:
1. Being liberal.
2. Rejecting orthodox Christian teachings.
3. Denying the doctrine of Biblical inerrancy.
These are three entirely different things.
I'm a Christian, I accept the historic teachings of the Christian faith as laid out in the historic Creeds, and confess that faith explicitly. I don't, however, subscribe to the doctrine of Biblical inerrancy; not because I doubt the credibility and reliability of Holy Scripture, I believe Scripture is credible, reliable, holy, and divinely inspired--it is the written word of God. One does not need to believe in inerrancy to believe in the inspiration and reliability of Scripture.
Further, I would also identify as "liberal", at least in the context of American national politics. I am left-of-center because I adhere to certain political positions that are usually identified as "left", these positions are positions I hold because I believe they are consistent with the ethics and teachings of Scripture and the Christian Church. I am not "liberal" in spite of being a Christian, I am "liberal" because I'm a Christian. It was Scripture and studying the teachings and theology of historic, orthodox Christianity that moved me to the left, I was raised quite conservative; but I found that I could not in good conscience be both conservative and Christian, it would have required entirely too much cognitive dissonance to do so. There are many who are Christian and conservative, and good for them, but as for me, I had to abide by conscience and the teachings of the faith--and that pushed me from right to left on many things.
What I mean when I say "I'm a Christian" is this:
I believe in one God, the Father Almighty, Maker of heaven and earth, of all things visible and invisible.
And in one Lord Jesus Christ, the only-begotten Son of God, begotten of the Father before all ages; God of God, Light of Light, very God of very God; begotten, not made, of one Being with the Father, by whom all things were made.
Who, for us men and for our salvation, came down from heaven, and was made flesh by the Holy Spirit of the Virgin Mary, and became man; and was crucified also for us under Pontius Pilate; He suffered and was buried; and the third day He rose again, according to the Scriptures; and ascended into heaven, and sits at the right hand of the Father; and He shall come again, with glory, to judge the living and the dead; whose kingdom shall have no end.
And I believe in the Holy Spirit, the Lord and Giver of Life; who proceeds from the Father [and the Son]; who with the Father and the Son together is worshiped and glorified; who spoke by the prophets.
And I believe in one holy catholic and apostolic Church. I acknowledge one baptism for the remission of sins; and I look forward to the resurrection of the dead, and the life of the Age to Come. Amen.
-CryptoLutheran