These are two very different things actually.
The Commander Planetoid (“deal XY% bonus elemental damage”) means that a second instance of damage will be added to your hit. For example, if you hit an enemy with a 1000 DMG melee strike, Commander Planetoid will hit the enemy another time with 750 DMG of the element it’s currently on. Technically this is two attacks hitting the enemy at the same time; if they’re different elements, you’ll see 2 numbers popping up – if they’re the same element, the game adds them together and shows just one number which is the sum of both.
I highlighted “bonus elemental damage” because this specific term denotes an important mechanic found all over the game. If weapons or shields or grenade mods have a “+50% bonus elemental damage” anointment, or if you have a skill that gives “bonus elemental damage” (e.g. Amara’s Forceful Expression), this is what’s happening.
“Bonus elemental damage” does not increase the damage of other elemental attacks – it just adds one more instance of damage on top.
The Elemental Projector (“Damage Boost”) is completely different. This doesn’t add another instance of damage, instead it increases the damage of all damage you deal with a given element.
So let’s use the first example again and say we’re dealing 1000 kinetic damage while we’re on fire. Since our attack isn’t fire, the Projector doesn’t do anything. Now if we do 1000 fire damage instead, the Projector boosts that by 1000 × (1 + 103%) = 2030 fire damage. It just doubled our damage. Unlike with “bonus elemental damage”, there is still only one instance of damage – but it is increased.
This works similar to Amara’s Tempest skill which generally increases all elemental damage, and in fact if you’re using both, they are added together: 1000 × (1 + 30% + 103%) = 2330 fire damage. The Elemental Damage guardian rank works the same.
Now in the case of the second artifact shown you can have both effects at once: Let’s say the Planetoid creates an extra “attack” with 750 fire damage while you are on fire. The Projector will then boost that attack to 750 × (1 + 103%) = 1523 fire damage. It will also boost your main attack if that’s fire too, it just takes any damage you deal and boosts it if it’s the right element.
Note that when I say “fire damage”, that’s base damage before it is modified again based on what you hit (flesh, shield, armor). More on that here.