(Note, I’m calling it the Decalogue to refer to the passages in Exodus and Deuteronomy without any preconceptions of how it’s split)
The Decalogue consists of 14 imperative statements which we are told belong to 10 commandments. (15 if you count “I am the Lord your God”) Obviously there’s a bit of a problem. We’re never actually told where those splits are. For the most part, though, we all agree on them. Everyone agrees that “Honor thy father and mother” is its own commandment, for instance. The difference is just in what number they’re assigned.
However, there are 5 imperative statements that people divide differently. Let’s look at the first 3:
Ex 20:2 “I am the Lord your God, who brought you out of Israel…” (Not actually one of the 14, but still relevant)
Ex 20:3 "You shall have no other gods before me"
Ex 20:4 "You shall not make for yourself a graven image…"
Ex 20:5 “You shall not bow down to them or serve them…”
There are 3 different ways people will group them. In Judaism, Catholicism, and Lutheranism, Ex 20:3-5 is all one commandment. However, we differ in that ALL Christians tack Ex 20:2 onto the first commandment, whereas Jews split it into its own. The argument in favor of this rendering is that to the ancient Israelites, idolatry and false gods were the same thing. You didn’t have graven images without worshiping them as false gods, and you didn’t worship false gods without making graven images.
And as you’ve probably guessed, the alternative is splitting it up with Ex 20:3 as its own commandment and Ex 20:4-5 as a second commandment. However, this isn’t unique to non-Lutheran Protestants. This is also how Eastern Orthodox churches do it.
The other 2 debated commandments are the two about coveting. And as the opposite to before, it’s Catholics and Lutherans that split them, and Jews, Orthodox, and non-Lutheran Protestants who lump them together into “Thou shalt not covet”.
Finally, there’s the topic of how they’re taught. Obviously it’s too difficult to teach the entire passages as the commandments. We want something short and simple to remember. It’s why the two-statement-long commandment starting “Keep holy the Sabbath” is shortened to “Keep holy the Sabbath” in catechesis. It’s easier to remember. So the confusion, then, that CARM and similar groups love spreading is that Catholics have completely removed the “graven images” bit, just because we shorten Ex 20:3-5 to “Thou shalt have no false gods” or similar. The fact of the matter is just that we see a difference (that the Ancient Israelites very well might have) between simply making a statue or picture of something and worshiping it as if it were God.
Source: The Division of the Ten Commandments
All Bible quotes from the RSV-2CE
EDIT: One minor detail I forgot to mention. Catholics and Lutherans order the two covet commandments differently. It depends on if you’re ordering based off Deuteronomy (Catholic) or Exodus (Lutheran).