You correct point out the difference between economic liberalism and social liberalism; but you don't extrapolate enough from it.
Economic liberalism is the only thing on the menu in the US, and in most other western countries at this point. Since the 1980s market liberal (or neoliberal) resurgence, no major political party, centre-left or centre-right, has really strayed from that line. So there is no point in distinguishing between Republicans and Democrats on that scale, because whilst the Republicans may be a little more "liberal" there, the Democrats have not by any means offering a socialist or paternal interventionist run economy. In recent years, some figures or new parties have finally stepped up to challenge that (Bernie Sanders in America, Jeremy Corbyn in Britain, Podemos in Spain, SYRIZA in Greece, etc etc), but by and large, economic liberalism is still the main offering on the table.
Social liberalism, on the other hand, has not been so dominant. Dating back to the civil rights movements; the Democrats got on the right side of history there, and were the liberal movement. The Republicans, by trying to conserve the status quo, became the Conservatives. Before the Civil Rights movement, the Republican party had always been known as the 'liberal' party, because of its economic liberalism. Since the 1980s, on Social Liberalism has really differed between the two though, and the Democrats got the tag of liberals from the earlier Civil Rights days, and have pushed that further with LGBT and minority rights.
Whilst you point out that the Republicans try to maintain freedom of guns, that's technically both liberal and conservative at the same time - liberal in freedom of choice, but conservative in maintaining the status quo (the definition of conservatism).