Example 1:
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Hypothesis 1: Robust being regular trait bonus
Pop bonuses = 20 + 15 + 10 + 10 + 20 + 5 = 80%
Multiplied by Special Bonuses = 5 + 20 = 25%
Giving us a total production bonus of = 1.80 x 1.25 = 2.25
With a base production of 8 Eng x 2.25 = 18 Eng [Screenshot says we produce 18.20 though]
Hypothesis 2: Robust counting as special bonus (like pacifism)
Pop bonuses = 20 + 15 + 10 + 10 + 20 = 75%
Multiplied by Planetary Bonus = 5+ 5 + 20 = 30%
Giving us a total production bonus of = 1.75 x 1.30 = 2.275
With a base production of 8 Eng x 2.275 = 18.2 Eng [Appears to match screenshot production]
The same applies to Gaia World 10% bonus production:
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It's a special bonus stacking with all other bonuses.
That's why Gaia World ascension perk is actually worth it, despite many people thinking otherwise. Because it multiplies all previous bonuses, just like pacifism/governor bonuses do.
And that's why Machine worlds perk sucks so much, because even though it gives +20%, it's a regular bonus like pop traits.
Once you hit +100% production bonus through traits, techs and civics, that +10% from gaia world allows you to produce more than +20% from machine worlds.
Simply converting machine worlds bonus from regular to special multiplier identical to gaia worlds, would be a badly needed buff for ME's and one that's completely justifiable.
PS: Pacifism tooltip is also flat out lying. It's not a 5% pop resource production bonus. It's a 5% production bonus on everything including stations.
Anyway, empire bonuses now affect (or may have always affected, and only megastructures is new) space resources. Adding a display is definitely new (if you have +50%, 2 looks like 3, and mousing over in system view shows you your modifiers).
So, on the wiki it says "empire", can't check what it says in game right now. Would not be surprised at all if some refer to it as an empire bonus, and some as an empire-wide pop bonus... In fact, I'd be more surprised if they're all consistent

Capacity overload, prod targets/omnifarious and, most impressively, instrument of desire covenant, all do the same. As well as the +5% energy prosperity tradition.
As for your math... Hmm... I agree. I don't see any other reasonable way to split those bonuses into a group of 75% and 30%, and that's the only way the math works.
Additionally, while I'd expected indenting and headers to mean something, like for intellectual, I'd also expect order to mean something, and "robust" is definitely conspicuously apart from all the other trait bonuses.
Well, anyway, when I get home I'll look in the game files real quick.
--------- long tangent incoming --------
That's silly about Gaia and machine worlds, but, if intentional, it does have one big impact. Driven assimilators. Before, once you had machine worlds, the only use for cyborgs was on habitats and ringworlds, where they got +30% to energy/minerals while robots would only get a pathetic +20% (and robots stuck to machine worlds where they had +45% everything). Another guy brought up that, with 2.0s nerfs andnchanges, while the machine pops get their reduced +35% to everything on machine worlds, cyborgs would get +40% energy/minerals, or +50% minerals if you found a species you could mod to indistrious/very strong. Which doesn't seem worth it.
If it's multiplicative vs machine world additive, and both those are intentional, then it could be. Of course, the new ME production tech doesn't apply to cyborgs, and if that's intentional, it's back to not being worth it. But, if it ends up applying to both... Making worlds with heavy mineral deposits into Gaia worlds, and then making the rest into research heavy machine worlds, sounds fun
Edit: but, in my experience, ME robots would have a reaaaal tough time reaching 100% additive bonuses without the +20% from machine worlds. The edicts affect space resources, so are empire bonuses, so I imagine they're in second multiplied group, and any addition to that group makes the first group stronger. Like, even in your example, you have pacifist and governor (and robust apparently) for +35%, and 70% pop bonuses. That means the 10% from Gaia world is +17% of base. But a 20% tile bonus (obviously not fromachine world) would be +27% of base.