Not exactly. Say that your character has Charisma with one skill feat gained in that skill.
Before gaining the Arcane:Charisma + 3, your Arcane is non-existent, i.e. 1d4. Once you gain Arcane:Charisma + 3, your Charisma will be still 1d8 + 1 (the one skill feat mentioned above) and your Arcane will be 1d8+1+3 = 1d8 + 4. The modifiers add up. Keep in mind that your Arcane check will be ALSO a Charisma-based check. So any things that add to Charisma checks will add to your Arcane checks.
If you do not gain the power feat, Arcane is 1d4 and it is not Charisma-based or any-other-skill-based. Just a plain 1d4 with no modifier.
Is this somewhat understandable?