I was lazy picking Wikipedia when everyone knows it's got an American brainrot problem. That's entirely my fault.

It is true that "conservative opposition to liberalism" is a thing that has exist and currently exists, but the issue is that "conservative" is a relative term, it refers not to an absolute ideological tendency (like liberalism does) but to the necessarily relative value of seeking to conserve the current order of things. This is relative because the order of things can be different, and that changes the question of if you want to conserve it (conservative), go back to some past state, real or imagined (reactionary), or advance to some future state of greater development (progressive).

So when liberal revolutionaries set the west on fire, conservatives were in conflict with them because the conservatives were trying to preserve the feudal/aristocratic/monarchic order that the liberals opposed. Now that the liberals in the west are no longer revolutionaries but overwhelmingly the establishment and without any serious contest, the acting of promoting liberalism over other ideologies is conservative and the old position of promoting a feudal/aristocratic/monarchic order is reactionary. The rise of neoliberalism, in particular, represents the overwhelming historical victory of liberalism over both reactionary and progressive forces ("There is no alternative," the perfect conservative slogan).

Of course, a political ideology can be a mix of conservative and reactionary or conservative and progressive (I'll let you decide on reactionary/progressive), and I'd say that former pair is pretty important for understanding the ideology of the Republicans, but don't let that exaggerate in your mind the piddling degree to which the latter pair applies to Democrats.

Is that a better explanation? Whether this is how you personally want to use the words or not, this will help you understand how socialists use them.