No, that person using today’s media-created vernacular is a “reactionary”, not a conservative. Which again is not a word I particularly care for.
I might also add that many non-conservatives on the right are also interested in creating moral and ethical principles that were never before codified, even if they were in common usage. For example, even though parts of the US have in past spoken in languages other than English, only today is the sensibility that making English the official language is desirable.
This is novel, and only “reactionary” in places where efforts a bilingualism have already intruded. In many places where English only is being advocated, local bilingualism was never much an issue to begin with.
But in neither case, is pursuing bilingualism or English as the official language on the agenda of true conservatives, as they are satisfied with the status quo of *neither* being on the books as law. They see no pressing need.
A far more serious example is abortion. Conservatives are generally uncomfortable with it, but accepting of some degree of it in their region. Some degree of the shades of gray between full legal abortion and no abortion. They are not “issue focused” enough to want to do anything about it.
The “reactionaries”, again disdaining that word, are far more absolutist in wanting to eliminate any “color of law” from abortion, at least at the federal level, and in their particular State as well.
But even among them there is argument about limitations: fully illegal, States rights, no federal funding, and even some libertarians who are pro-abortion.
What they all have in common is that they want strong *change* to what exists today. True conservatives either want little change or gradual change.
Using that meaning for “conservative”, makes everyone into a reactionary. I’ve never met anyone who wanted everything to stay just as it is.