Liberal republic is a republic which has proclaimed ideas of liberalism as the core foundation and reason of its existance. This could be defined by constitution, declaration of independence or another charter document. It seems strange nowadays but some liberal republics had no constitution.
Speaking about liberal republics, there is an interesting reading:
http://belfercenter.ksg.harvard.edu/files/fischer.pdf
P.S. I agree that these voting options are not so simple. This was made intentionally in order to get more precise figures of political preference.
Thanks for your welcome. I understand that a liberal republic is a place where the government doesn't interfere in anyone's daily life, and the US shall be regarded as such, but the problem is how to fix a limit on what the government of a liberal republic can do. It's dramatic that to start most kinds of business nowadays, one needs a licence and that on any income this business will generate, one must fill forms about it and share part of it (in some liberal republics a large part) with the liberal republic's managers, who may sometimes redistribute some of that.
I should be a partisan of liberal republics, but as a business owner (well, very small business), I often dream of anarchy.
The US could only be a liberal republic by your definition if it did not interfere in ones daily life. Are you saying the US doesn't interfere in ones daily life?