What is the damage dealt by Eldritch Smite?

Eldritch Smite is an Eldritch Invocation for Warlocks similar to the Divine Smite feature from Paladins. It is available in Xanathar’s Guide to Everything and states

Once per turn when you hit a creature with your pact weapon, you can expend a warlock spell slot to deal an extra 1d8 force damage to the target, plus another 1d8 per level of the spell slot, and you can knock the target prone if it is Huge or smaller.

So, let us say a 5th level Warlock (single class) uses its 3rd level spell slot to proc Eldritch Smite on an enemy they hit.

What damage do they roll?

I am asking this because a Warlock that I have been playing with recently (we are running Dungeon of the Mad Mage) has been rolling 3d8, but my reading is that it should be 4d8.

To be clear on the reasoning, these are the two interpretations clashing:

  1. Casting it as 1st level is 1d8, and any extra level is +1d8. This gives 3d8.
  2. Casting it naturally deals 1d8, independent of spell slot. Then another (level)d8 dice from the spell slot. This gives 4d8.

Ultimately I find the wording confusing. The first interpretation is in line with how spells are usually described, but the second is what I understood when reading it. I figured I would ask before telling the other player he has been rolling wrong for weeks (and maybe being wrong myself when doing so).

For comparison, Divine Smite is way more precise in its wording:

Starting at 2nd Level, when you hit a creature with a melee weapon attack, you can expend one spell slot to deal radiant damage to the target, in addition to the weapon’s damage. The extra damage is 2d8 for a 1st-level spell slot, plus 1d8 for each spell level higher than 1st […]

To be clear on my expectations on the answers: either a compelling and careful argument on the reading, or any further clarification from any official source, even as simple as an example of damage calculation where it is clear which damage was rolled.

Related: This answer seems to be using my interpretation in its math, as well as this. This answer also seems to be using the second interpretation (4th level spell slot = 5d8 damage). Finally, this question also assumes the second interpretation.