PRO LT Handlebar Stem asks to tighten top handlebar screws first before bottom screws? Conflicting manual instructions?

It is an 'eternal' and more of a philosophic question. Both variants are out and both seem to be true and not so. It is the same on the motorcycle front. The picture of the actual stem says clearly that you should close the gap at the top. I would follow that procedure. Screw the bolt in just until the plate touches the opposite side, hand tight, not with the extra quarter of turn. Then tighten the bottom plate to the recommended torque. That way the top bolts, the plate and the part of the stem behind the bar will act as a single element and the plate plus the bolts act as elastic elements.

If you tighten them cross-wise, the bar is compressed into an oval shape and has to take up more of the clamping forces.

(This is merely an opinion, but I think it is mathematically impossible to evenly tighten 4 bolts crosswise with the same torque. Five bolts, like on the wheel of a car, yes, but not 4! Bolt 1, torqued, bolt 3, torqued, but then bolt 2 or 4 are breaking the crosswise rule! Whereas with 5 bolts it's 1-3-5-2-4.)

You should follow the instructions on the yellow cardboard. The manual contains the normal default procedure of tightening crosswise and evenly, but the second bullet point, "In case of additional instruction is added between stem body and stem cap, follow this instructions carefully." is telling you to override this normal procedure if other instructions are given.

The yellow cardboard is giving you these override instructions, which also match the markings on the stem itself for additional certainty ("close gap").

The card is telling you to tighten the top bolts until the gap is closed, then tighten the bottom ones. It does not mean to torque the top bolts to final tightness before tightening the bottom ones.

As others have alluded, the instructions do not say completely tighten a bolt all at once. I interpret the yellow additional instruction card to mean "make sure there is no gap for the top two bolts; there may or may not be a gap for the bottom two bolts."

To do this without letting things get unbalanced, I'd follow the "Z" cross-tightening pattern, turning each bolt a 1/2 turn at a time initially. Once the cap gets close to being seated, turn each bolt only a 1/4 turn at a time. This way, the 2 top bolts should seat first, ensuring the no-gap on top. Continue to the torque spec in the Z pattern with 1/4 or 1/8 at a time turns.

Here are a few diagrams I drew in Autodesk Fusion 360 to explain how these zero-gap stems work.

According to the instructions provided with the few zero-gap stems I've encountered, you should tighten the "no gap" bolts to full torque first, then alternate between the lower bolts to fasten the handlebar. That makes sense according to the diagrams because that ensures the upper tabs are solidly clamped together to form the C-shaped clamp. If you're confused by my weird terminology, take a look at the diagrams (especially #3 and #4). You'll see what I'm talking about.

This is a normal stem. The left piece (the stem's faceplate) evenly tightens in a straight line towards the right piece. Assuming it gets properly torqued, the faceplate will travel in a straight line to the right as the bolts are tightened.

Next, this is what would happen if you tried to use a normal stem in a no-gap fashion. As you can see, the two pieces are no longer parallel, and the bolt holes are no longer aligned properly. That's not good. If you were to tighten the bolts while the stem was arranged like this, you run the risk of damaging your stem and handlebars because the faceplate will be forcibly bent into an oval shape so that the tabs (and thus bolts) are parallel once again. Obviously, you cannot clamp a round bar with an oval-shaped clamp.

The next picture represents a zero-gap stem installed correctly. Note how the upper tabs are flush with each other and are lined up parallel; you're effectively turning the two separate pieces into one solid C-shaped clamp. When you tighten the lower bolt, the top joint will stay still. Instead of the faceplate moving in a straight line towards the right like how it would with a normal stem, it is now deflecting in a circular manner.

And finally, this is what it would look like if you tried to use a no-gap stem like a normal stem. You can see how the tabs are completely out of alignment. Like before, the faceplate might get deformed into an oval shape if you try to tighten it down.