For one example, a constitutional conservative may also believe in Christianity and therefore an unfamiliar liberal person may free associate and have knee jerk reaction to believe that the conservative wants to combine the church and state.
A constitutional conservative would want a powerless federal government and abhor government enforcement of theocracy. Oppositely, a religious conservative would want a government that is powerful enough to use those powers of federal government for the enforced values learned in religion.
It serves the purpose of liberals and progressives to cloud the conventional wisdom that defines a US Conservative.
A US Conservative speaks of the Federal Government in terms of what "they" are doing. A US Liberal spaeks of the Federal Government in terms of what "we" are doing.
A US Conservative thinks in terms of Federal Government. A US Liberal thinks in terms of National Government
A US Conservative thinks in terms of State and State Sovereignty. A US Liberal thinks in terms of Provinces and National Soveriegnty.
A US Conservative believes the individual gets power from God and the individual gives that power to government. A US Liberal believes the individual gets power from the government and there is no other source of power.
A US Conservative believes government has limited powers only within the definition of the Constitution. A US Liberal believes the individual has limited powers only within the judgement of the all-powerful government.