the reason for the expression born-again is because the Bible / Jesus are clear that entry to the Kingdom of Heaven is not through how you are born (huge issues in the day with Jews assuming that they were born into the Kingdom of Heaven), but through making a deliberate and personal choice to be born again into God's Kingdom...
The reason for the distinction now is that we live in a country where the word Christian has lots of meanings, and in particular an assumption that you are a Christian if you are not anything else - partly the fault of an established Church which is built into society... However the Bible is very clear, you can not be born a Christian - it is a choice, so the terminology is used to distinguish those who make the choice v. those who make an inaccurate assumption that they are already a Christian and need take no personal responsibility for accepting Jesus' death / salvation.
So, I would agree with others, it is not a term I would use about myself by default, but equally it is a term I am comfortable with...
The issue about feeling comfortable is a difficult one - in your original comment you say I am a very private person and like to pray and worship quietly in church but enjoy the feeling of being lifted up by the sermon and the Bible study. This is an area the Church has struggled with over the years - understanding when someone's character means that they find certain approaches easier (e.g. introvert preferring one style of worship, extrovert another) versus some people who say the same as a defence mechanism against having to actually commit as Christians.
I think for me it is very clear - Christ doesn't require us all to worship him / God in the same way, and there is certainly no prescribed form of Church / being a Christian, so you are not wrong in what makes you comfortable... however equally, we are called as Christians to do various things which may not sit comfortably with you - to be outward in our expressions of worship / evangelism / etc. is one simple example and we do have an issue in this country of not being very free in worship, of finding reasons and barriers to avoid having to actually worship God - using our attendance at Church as an excuse to not actually have to engage in a daily relationship with Christ... Looking at Asian & African churches currently gives a very good illustration of where God is being allowed to work - perhaps if we had a culture which embraced God as a part of our daily life rather than just put in a box at 10am on a Sunday where we sing a few songs and say a few prayers and listen to a homily - then we would see God working more in our society which perhaps might not be a bad thing... (certainly a personal challenge...)
For each of us it is a very personal journey, but I think it is worth exploring why you feel uncomfortable, and how that sits in your personal journey with God - something that is unique to you - so not right or wrong, but perhaps opens lots of good and challenging questions?