as for humboldt/standard/birdsmouth....
all of them direct just as well as the other, each has its use, really the hinge/hold wood does the directing, the face cuts whichever type simply provide room for the stem to move.
Standard is good if you are in open ground and would like a very low stump. Its also arguably the easiest to master, and therefore the most common, though I do see a whole bunch of folks taking ridiculous amounts out of the stem for what amounts to a very shallow face... a 45 degree opening is more then ample for 99% of the wood out there, yet you see many that are nearly 90 deg essentially just cutting the taper out of the stump...
Humboldt is good for getting the butt to hit first, as well as acting as a chock against back slip, and will allow a tree to "fly" especially if its a narrow face down hill, and if you want to do crazy stuff like soft dutch, or siswheel it works marginally better then standard, and it doesn't ruin marketable wood. There are a bunch of other things you can do with a humboldt too, like sniping, side shift, modified block face etc etc etc.
Birdsmouth is good if you have brittle short fiber wood and you want to be sure it goes where you intend, also handy in chair prone timber, though the effect is marginal. Essentially, its used for when you absolutely don't want the face to close up and stall thereby shearing the hold wood, or causing a chair.
A forth unmentioned would be a block face, which is were the humboldt originated from, you take a full on square chunk out rather then a sloping cut, its really a lot of work to make happen so its rarely used unless on very big timber or if you want the butt to sit down on the stump and not tear up ground in front of said stump, can be handy for falling over paved or sensitive areas, though a modified humboldt, which is a normal humboldyt with a fat chunk takin out of the leading edge is a whole lot less work, and you don't need to be as accurate with the block cuts to get the same results