I agree with your interpretation of McClellan's order - it was another poster who seemed to be suggesting that Burnside complied with McClellan's order simply by taking the bridge.
I'm not saying McClellan complied with Burnside's order
simply by taking the bridge - I'm saying that for McClellan to show substantively that Burnside
disobeyed his orders it is not sufficient to show that he is carrying them out a bit slowly (so long as he is in fact engaged with carrying them out).
If Burnside is arrested at 10 AM, or 11 AM, or noon, then in subsequent defence he can argue that he
was following orders - he'd had to fight for the bridge, but he wasn't ignoring it and was pushing troops forwards. It wasn't possible to take the bridge
quickly - or so he could argue - and disobedience of a lawful order (a death penalty offence) doesn't and shouldn't cover a situation where you're assigned a difficult task and haven't completed it straight off - so long as you're engaged with doing it, or have reached a point you can no longer do it.*
This is why Key was sent with the last-chance order - my understanding is that at that point it's "charge
now or you're arrested". If Burnside is being dilatory about
that (and doesn't charge) then he's clearly not obeying orders (it's open and shut); if he charges straight away then either he succeeds (and that's been accomplished) or he fails (and it was an impossible task, and arresting someone for failing at an impossible task is obviously not logical).
Certainly Key could have been sent at 11AM with the last-chance order - that might have made it work out, or it might have got Burnside arrested. 10AM is too soon because at that point Burnside has done "all he can do" (as in, the only regiment in position has charged and been repulsed, the next units are just reaching it and the bridge will then fend off two more regimental charges.)
Once the bridge is taken, the next step of the order to be followed is to push troops over the bridge, up onto the heights. There is, and has to be, some discretion in this one because of the old relationship between speed and mass - if Burnside pushes troops up onto the heights immediately then he's got a column of fours advancing with no weight or batteries behind it, which would be easily defeated. (This by the way is the same reason the Prussian relief column at Waterloo is delayed after crossing their stream, they need enough force on the far side of the river to advance a balanced force that can't just be crushed.)
Burnside thus (legitimately) needs to collect some troops on the far side of the Antietam before pushing up to gain the heights; the question is
how many troops, which is why another order is sent over when McClellan thinks Burnside should have begun pushing troops up on to the heights by now.
* for example, if Commander Pseudonym is ordered to take a fortified enemy position and charges it and is repulsed, he has clearly followed the order even if he hasn't actually taken the position; certainly he doesn't deserve to be arrested over it. If Commander Pseudonym is ordered to take a fortified enemy position and begins moving troops around the flank to render the position untenable, then you can't take a snapshot half an hour into the process and say Commander Pseudonym is disobeying orders because he
hasn't charged yet. This is because most orders have some latitude to them, as a commanding officer is meant to show at least some initiative.
If on the other hand Commander Pseudonym is ordered to take a fortified enemy position and does nothing, or if Commander Pseudonym is ordered to
charge now and
doesn't, he
has disobeyed orders.