To reiterate what I said in the original article, there are no right or wrong answers to this - the ideas below are what I would do in the situation, but it's entirely possible that there are better choices, or even that the these ones would backfire completely! Also as someone commented on reddit, all of this is '100% subject to change based on any one of a thousand shenanigans' - little details like positions on the field, luck, or pilot mistakes can make all the difference, and no plan ever survives contact with the enemy.
You: 10 mixed t1 cruisers with frig support
Opponent: 3 drakes, 3 caracals, 1 raven, 1 blackbird
I originally included this one as an easy example, since I'd specifically mentioned drakes and caracals in the original article. I then decided to add the raven and blackbird to make it a little more interesting, and in retrospect that made it a very challenging fight (quite a few people have suggested they wouldn't take it). Let's assume for the purposes of the example that we're feeling luck and we're going to take it.
The choice between drakes and caracals is fairly simple; the drakes are a greater threat, but the caracals are much more fragile (incidentally, assault missile caracals would also be a greater threat to our frig support). Because of that, killing the caracals first will reduce enemy threat quicker, and they should probably be primaried early. The raven does more dps than the drakes and doesn't have that much of a better tank, however as many of you rightly said you can afford to leave it until last. The reason for this is that while its damage output is stronger, cruise missiles or torps do significantly reduced damage against cruisers and frigates, and as a result the raven presents relatively little threat. Obviously if the blackbird is in range then it should be primaried immediately - it presents a huge tactical threat to t1 cruisers and frigs with their low sensor strength, however the chances are it will be at long range. While actually catching it may prove a challenge, sending a few fast frigates its way should force it off the field, suppressing the threat (or if the pilot is slow, get it tackled and provide you with a kill).
You: 20 t2 frigates (mixed assault ships and inties)
Opponent: 3 zealots, 1 curse, 2 purifiers, 1 abaddon, 1 heretic
The aim of this example was to demonstrate how threat can vary based on your gang composition. In this example the main dps ships are fairly ineffective against frigates (abaddons and purifiers due to their large weapon systems, and zealots due to a lack of a drone bay and mid slots for webs etc) while the curse, which has a fairly low damage output, is incredibly dangerous due to its bonused drones and long-range medium neuts.
The heretic presents little direct threat, but getting rid of their bubbler will make it much easier for your frigs to warp in and out of the fight when they are endangered. It's also fairly fragile, so I'd probably just kill it first off while I take in the rest of the situation. The curse needs to die as soon as possible, and the outcome of the fight really depends on how many of our gang are left alive by the time it goes down. The purifiers are a good example of a where pilots should take their own initiative - they're so fragile that there's little point primarying them, but if one is decloaked close to somebody they should just grab it and pop it. With the curse down, the zealots with their medium guns are the biggest danger followed by the notoriously tanky abaddon.
You: 3 mid-range battleships (e.g. pulse abaddons or AC tempests), 1 arazu (point and damps)
Opponent: 4 hurricanes, 1 broadsword, 2 jaguars, 1 stiletto, 2 scimitars
This is my favourite example, and is based on an actual situation I've found myself in (admittedly we were flying another battlecruiser gang rather than BS, but the concept is the same). What makes this one interesting is that it's an example of a situation where it can actually pay to go against your instinct - the instinct which tells you to primary the enemy logistics ships.
Of the 10 man gang opposing you, only 4 ships (the hurricanes) pose a direct threat. Everything else is a tactical threat, which makes it harder for you to win the fight but presents no actual danger itself if the direct threats are destroyed. Not only that, but the relatively fragile shield hurricanes are likely to be much closer than the fast, kiting scimitars - close enough to hit with large pulses and ACs with full damage. In this situation, I would primary the hurricanes immediately while your dps is still in-tact, and if possible put range damps and any ecm drones you have on the scimitars. This is a gamble - with three battleships you should be able to punch through the reps of one or possibly both scimitars, but once you lose a battleship (which may be first or second depending on whether your opponent chooses to primary the arazu) this will become much more challenging. Assuming you survive long enough to kill the hurricanes, you'll probably find the opposing gang will disengage, although you may be able to snag the broadsword or pop some tackle as they do.
The alternative strategy would be again to range damp the scimitars, and then to switch primary between the hurricanes (bringing the scimmys in close to apply reps) and the scimitars once they are in range. However due to the manoeuvrability of the scimitars, you may find they continue to evade you and you're unable to kill anything at all, which would be very embarrassing!
You: 5 kiting battlecruisers
Opponent: 2 megathrons, 2 vexors, 2 brutix, 2 enyos
I admit I'm not a huge fan of this example; I'd done the three I wanted to cover and mainly just included this one for completeness. That said, there are still some useful things we can say about it.
In this situation, all the opposing ships are very dangerous up close - if their gang were to get on top of you, you would almost definitely lose. However, your gang is much more manoeuvrable and can out-run their main dps ships quite easily so long as you aren't tackled. To this end, I'd primary the opposing ships roughly in order of speed - enyos and vexors first (the only ships faster than your own), followed by brutixes and finally the megas. Once the faster ships are down this one should be a fairly easy fight, but sometimes that's just how it goes.