I added it twice on purpose. Improved Smite on PHB 85 is explicit that you add the extra d8 to both your regular attack and your smites.
Yeah, I think you are misinterpreting this.
"Whenever you hit a creature with a melee weapon, the creature takes an extra 1d8 radiant damage. If you also use your Divine Smite with an attack, you add this damage to the extra damage of your Divine Smite."
It doesn't state that you add this damage a second time. That isn't even a literal interpretation (but I could see someone interpreting it that way). It says (and probably means) that the extra D8 is added to the extra damage of the Divine Smite. Period. All of that radiant damage is one pool of damage. In other words, since the name of this is Improved Divine Smite, the 1D8 extra damage from IDS stacks with DS, it does not get replaced by it. A first level spell slot adds 3D8 (adding it once), not 2D8 (not adding it) and not 4D8 (adding it twice).
I think the intent was to avoid confusion on whether the 1D8 applied for an actual DS attack or got replaced by it, and they ended up making it less clear.