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The difference in the rounds are # of shots, AP and Damage. I know that there is one, but I am struggling to put together the logic map for which round is best suited for which type of character. As a start, I imagine that having more shots, Heavy D3, with add 2 to the hits roll but no AP, (unless in the Devastator Doctrine phase then AP-1) is best against characters that have low Save values or a 3+ or 4+ invul because you just want more shots.

Did some math versus various targets for you:

I assumed that the eliminators were past devastator doctrine, did not have any aura buffs or enemy unit debuffs, and weren't Raven Guard.
6 targets: Company Commander (T3/5+/5++), CC in Cover (T3/4+/5++), Space Marine Captain (T4/3+/4++), SMC in cover (T4/2+/4++), Artificer Armor/Stormshield Smash Captain (T4/2+/3++), and Smash Captain in cover (T4/1+/3++)

Labeled as 1A/1B, 2A/2B, 3A/3B respectively

TL/DR:
Mortis rounds beat out everything else handedly. they have AP-2 which on most characters is enough to get to their invuln save even in cover, making the ignores cover part of executioner rounds pointless. Then on top of all of that they have d3 damage and possible mortal wounds to go past invulnerable saves entirely.

maybe there would be a better choice for certain larger targets, but as far as I can tell in every possible circumstance you should never be shooting anything but mortis rounds at characters. Once all the characters are gone then hyperfrag is useful to try and squeeze some more anti-infantry shooting out of them, but again mortis rounds would be better suited for vehicles and monster hunting. executioner rounds might be better against "always in cover" units like Jormungandr Tyranid Warriors who would ignore AP-1/-2 from their adaptive physiology, so that you essentially still get a -1AP from ignoring cover in them opposed to nothing from mortis, BUT MORTIS STILL WINS FROM THE BONUS WOUNDS CAUSED.


Executioner rounds:

1A:
3 shots
2.5 hits
1.666 wounds
1.111 unsaved
1.111 damage suffered
1B:
Same

2A:
3 shots
2.5 hits
1.666 wounds
.8333 unsaved
.8333 damage
2B:
Same

3A:
3 shots
2.5 hits
1.666 wounds
.555 unsaved
.555 damage
3B:
Same


Hyperfrag rounds:

1A:
6 shots average
4 hits
2.666 wounds
1.777 unsaved
1.777 damage
1B:
Same until saves
1.333 unsaved
1.333 damage

2A:
Same until saves
.888 unsaved
.888 damage
2B:
Same until saves
.444 unsaved
.444 damage

3A:
Same until saves
.444 unsaved
.444 damage
3B:
Same until saves
.444 unsaved
.444 damage


Mortis rounds:

1A:
3 shots
2 hits
1.333 wounds
.888 unsaved
1.777 damage
+.222 mortal wound damage
1.999 total damage
1B:
Same

2A:
Same until saves
.666 unsaved
1.333 damage
+.222 mortal wound damage
1.555 total damage
2B:
Same

3A:
Same until saves
.444 unsaved
.888 damage
+.222 mortal wound damage
1 total damage
3B:
Same

Yes I also ran some extensive simulation to test against different targets and strategies. (Mostly I was also testing to see if it's worth dropping the shot from the sergeant to get the +1hit and wound.)
I found that Mortis is the best vs nearly all targets. And the dropping the sergeants shot is usually best in order to get more chance of the MW. (Only times when it seemed not worth it was if your already getting a +1to hit and wound from raven guard).
Also master artisans works well, even if you successfully wound, still go for the reroll to try and get the MW.

In game I have always been disappointed with hyperfrag.

Mortis rounds are absolutely amazing. I played at a tournament recently and the Mortis rounds while using guided aim was extremely effective. Notable kills were a Vindicate assassin, Drukari warlord and the creme de la creme was taking the final wounds off a stomps (rolling a 6 to wound causing mortal wound then getting 3 on the damage roll) which then caused a 9 inch explosion destroying a deffkopta, 2 trukks and a bigmek.

I'm including a third squad in my list from now on for sure.

Mortis are the best for sure, but if your opponent hides characters out of LOS at least we have the Executioner rounds to still put some damage on them.

I love Eliminators and run 3 units in almost every list I play. As Dark Angels, they love the reroll 1's to hit when they stand still, and getting and extra 6" on the first turn (from DA's super Devastator doctrine) is the cherry on top.