Does a 10W or 12W charger make an iPhone willing to boot up sooner? I cannot be entirely sure about this but it seemed to happen this way:
When the iPhone 5, 6, or 7 was out of power, I could plug it into my car's USB charger or a portable USB battery charger for 10, even 15 minutes, before the phone is willing to start up.  A 15 minute wait, can be a bit too much -- if it were 2, 3 minutes, it would be more tolerable, but a good 15 minute, is too much.
But I noticed sometimes when I plug in the lightning jack for a charge, the iPhone will start immediately.
Is it due to some charger are supplying 5W, while some are supplying 10 or 12W?
I suspect iPhone might be doing some calculation: if possible power consumption is less than power input (10W), then be worry free and power up immediately. However, if possible power consumption is more than the input (5W), then don't power up first, accumulate first before powering up (and it turned out to be 15 minutes usually).
Does someone know for sure how this works?
P.S. Decemeber 2016: So today, I tried out an iPhone 5S, and saw that it was 1%, but too late, about 1-2 seconds after I plugged it in, it turned itself off. And thinking that it was a 39W 2 port charger with QC 3.0 (an Anker charger used in the car), I thought it should start up almost immediately. Not so, it took close to 15 minutes before it turned itself on, and I didn't look what battery level it was when it started up, but 10 minutes or so after the start up, I looked, and it was at 21%. So it looked like it could have turned itself off sooner.
Many factors - dead phone/0% battery isn't an entirely fixed point, depending on battery age/health, power requirements, temperature (winter cold...), calibration state etc it can mean anything from a calculated shutdown by the device at a level before excessive wear starts to occur, to the phone just suddenly dying at a point where there's still supposed to be juice left and it's not so much protecting itself, more like a blackout.
During discharge the voltage continuously decreases, which is what the phone keeps tabs on and cuts off at a safe level, but as it loses charge the max amperage/wattage also gives way, leading to further (short) voltage drops when it overextends itself.

As an example, if you're running low on battery and decide to take a bunch of photos before your phone dies, the high power draw will likely mean the phone shuts off a lot sooner than if you kept it in your pocket on standby.
But that means that in the latter scenario... once it's dead, it'll be deader, and it will take longer to bring up to a healthy enough voltage to boot - turning on, mind you, is a lot more power intensive than just staying alive once on. If the phone tries to boot before enough power is available it will black out again.

Input power obviously plays a large role here, just as you say, but it's not so much the phone calculating anything fancy or being tactical. Plug an iPhone with a worn-out battery into a semi-glitchy cable and it will happily bootloop for weeks on end instead of waiting and building up enough charge to actually manage to power on and chill out. 

You're on the right track, key thing being that the brick can't power the phone directly, it's feeding the battery which feeds the phone, again a battery at low charge is flaky so even plugged in there needs to be a buffer, and once you get going that where you get the non-linear response. Say the phone needs 2W and it's getting 4W effective from a 5W brick, so 2W left to build charge. Double it to 8W and the surplus triples.