“the First Commandment doesn’t forbid the making of images; it forbids making them for the purpose of worshipping them”
No, there’s two sentences which both contain a prohibition on a different thing. Both are forbidden.
If I said to my kid: “Don’t borrow the car without my permission. Don’t return the car without filling up the gas tank.”, then nobody would assume I meant the kid could borrow the car without my permission as long as he filled up the gas tank.
No, there is no punctuation in the original Hebrew; it is a translators choice. Likewise, even the numbering of the commandments came into practice and the verse numberings were added much later than that. The command to have no other gods and to make statues to worship them is clearly one commandment.
They're not very different things. The first is a prohibition on worshipping a god other than the Lord. The second is a prohibition on making or worshipping an idol of such a god. The first encompasses the second; if you can't worship another god, you certainly can't worship an idol of another god.