No. This means exactly what it says. Many monsters have a feature called Innate Spellcasting. For example, the Storm Giant has:

Innate Spellcasting. The giant’s innate spellcasting ability is Charisma (spell save DC 17). It can innately cast the following spells, requiring no material components:

Your spellcasting modifier for those spells you cast using Magic Initiate is Charisma

You quoted the relevant text:

Your spellcasting ability for these spells depends on the class you chose : Charisma for warlock

So your DCs, to hit modifiers, etc. for those spells are all based on your Charisma, so: DC 16.

If you later learn those same cantrips from another source (multiclassing, Arcane Trickster roguish archetype) you can choose which ability score to use, if that second source uses a different ability score for casting. If you have the Arcane Trickster archetype, then your spellcasting ability for your rogue spells is Intelligence.