In my opinion, being liberal is being able to adjust with the times while remaining faithful to your core values and principles. Being liberal does not mean that you shed your own values and principles and follow whatever is trendy or "western." I find that to be fickle and insecure, not liberal.
It is quite possible to take pride and interest in one's culture and remain faithful to one's cultural values while adjusting with the times, so I don't think liberal and culture are contradictory. There are certain aspects of certain cultures that are not appropriate for every environment or situation but that can be managed with common sense and taking situational appropriateness into account. It is not necessary that people discard their entire cultural beliefs.
When a person is bound by 'core values and principles', isn't liberated from constraints like these, i wouldn't call that a liberal.
There are varying degrees of liberalism. What one person considers liberal, another may not.
As far as constraints, is anyone really free of all constraints? As much as some people may delude themselves otherwise, everyone has some sort of constraint, be it legal, social, moral, or personal.
Liberal yet faithfully religious. I learnt a long time ago that being conservative and being religious were two different things. So many people still mistake the two.