Think of vertically bi-amping as turning the amp on its' side i.e. so that it standing vertically on the shelf or next to the speaker rather than laying down ( as it normally would ) horizontally. Once you get the picture of the amp standing vertically, it will remind you that you are running one channel for the lower section of the speaker and the other channel for the upper section of the same speaker.
To further imbed this easy to remember idea in your head, think of how your speakers are spread out horizontally ( or side to side ) in the room. Horizontal bi-amping does the same thing i.e. uses one channel for the lows ( or highs ) of one speaker and then spreads the other channel of the amp over to the other ( horizontal ) side of the room for the equivalent section of the other speaker.
There are benefits to each type and which works best will depend on the type of speaker load that you have. If one is running a speaker that has a highly reactive mid / tweeter section ( E-stat's, etc... ) with multiple woofers, my opinion based on personal experimentation is that horizontal bi-amping works best. Since both sections of the speaker are going to pull pretty hard on the amps power supply, which is common to both channels in most designs, both the bass and treble may "suffer" at the same time. Having a dedicated amp for the treble with a separate amp for the bass helps to ease the load on each amp and power supply. Since it is unlikely that both frequency ranges would be "surging" at the same time, effectively narrowing the frequency range that each amplifier / power supply must feed can increase efficiency and minimize the stress on teh system as a whole.
If one has a "normal" speaker with relatively mild loads like cone mids, dome tweeters, and a woofer(s), which woofers always need quite a bit of power and have larger amounts of reactance / reflected EMF to deal with, one is typically better off with vertical bi-amping. This allows the power supply to shift most of the current draw to where it is needed with the woofer on one channel while it pretty much "coasts" with the channel feeding the mid / tweeter section of the speaker.
Obviously, these are generalizations and each situation is different, so you'll have to do some trial and error and see what you think works best. The one advantage to vertical bi-amping is that there is less potential for crosstalk to occur from the left to right channel than if both channels were being used in a normal "stereo" fashion. This is because, in effect, you now have "two channel monoblocks" feeding each speaker.
Having said all of that, i can understand the confusion between the vertical and horizontal nomenclature that we use. I think that this comes from the fact that most systems are installed in some type of rack with the equipment arranged in a vertical array. With this configuration, it is easy to assume that "vertical bi-amping" would make use of the two left channels of the amps that are stacked on top of each other for one speaker and the two right channels of the amplifiers feeding the right channels. This would use the two channels that were "vertical" to each other, making this a logical conclusion. The other method, using the two channels of one amp, which are typically laid out side by side, would be the "horizontal" mode of bi-amping according to this type of logic. Too bad "logic" doesn't always work with audiophilia : ) Sean
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