But it doesn't mean that you actually renounce citizenship of your home country.
Well from the linked Wikipedia page it does mean that you actually renounce citizenship. It's pretty clear in having you state that is the case. If you actually don't then you are lying.
>"I hereby declare, on oath, that I absolutely and entirely renounce and abjure all allegiance and fidelity to any foreign prince, potentate, state, or sovereignty of whom or which I have heretofore been a subject or citizen;
For example if your former country required you to submit a form renouncing citizenship / sovereignty then taking the Oath and not having done that seems a highly dishonourable thing to do (ie IMO you're lying).
From that Oath (granted my first viewing of it in full) it seems that no truthful person could become a citizen of both America and some other place unless they were American first and no longer made that Oath on any occasion.