Hi Ivan -
You're right - it does get confusing. I think the problem stems from the fact that the term "bi-amping" is pretty generic, since all it means is using 2 amps, and there are quite a few options for how it can be done. In its original form, bi-amping (or tri-amping, etc.) involved the use of an electronic crossover to split the signal before the amplifiers, and the amps were connected directly to the drivers. More recently, speakers have begun to offer multiple inputs to the built-in crossovers to allow for bi-wiring, but this also opens the possibility of using multiple amps. This is still technically "bi-amping," even though no external, low-level electronic crossover is used. Both amps are driven full-range, and the speaker's own internal crossover does the job of shaping the signal sent to the drivers. There can still be a real advantage because multiple amplifiers can deliver more power with lower distortion, etc. However, you open a major can of worms at the same time if you use different amps, cable, etc.
I have experimented with various forms of bi-amping, and I can offer a few observations:
1) "Passive" bi-amping (which does not use an external crossover) can work very well if it is done with identical amplifiers and speaker cable. It *may* work well with different amps, but this is an extremely complex issue.
2) Passive bi-amping may be arranged either "vertically" or "horizontally." I believe the vertical arrangement works best, with one matched stereo amp per speaker and matching speaker cable. Each amplifier is driven with one channel's full-range signal, sent to both inputs (using a "Y" splitter across the inputs), and each output drives a separate section of the speaker. This means you are not only bi-amping, you are effectively running the amps as 2-channel mono, so you've got a bi-amped, bi-wired, dual-mono arrangement. The horizontal arrangement assigns one stereo amp to the high-end of *both* speakers, and another stereo amp to the low end of both. Each stereo amplifier receives the usual L-R full-range stereo feed from the preamp, but one amp is connected to only the high-pass section of both speakiers, and the other amp is connected to the low-pass sections. This will certainly work, but it defeats one of the main advantages of the vertical option: when both channels of a stereo amp are driven with the same signal, the amp is effectively used in mono, and there is no difference signal (or crosstalk) across the power supply and ground system.
3) In my experience, the main reason people choose to try any of these methods is because they think they can combine the best characteristics of *different* amplifiers (and often different cable) for improved overall performance. While this may seem like a reasonable idea, in practice it is extremely complicated and rarely works as well as simply driving the speaker full range with a single good amp. The reason is because speaker design has become very sophisticated, with the drivers, crossover, wiring, and cabinet tightly integrated into a seamless whole. Furthermore, many bi-wireable, full-range speakers have the low-to-mid/high crossover point somewhere in what is effectively the midrange, where our hearing is most sensitive. It doesn't take much change to upset the coherence of these designs. So, whether or not this sort of bi-amping works and, possibly, works well depends a lot on the speaker design and where its crossover points lie. I know a number of speaker designers who have chosen to sidestep the whole issue by eliminating bi-wire inputs. That way, it can't be screwed-up.
4) Using an external, electronic crossover does not necessarily help. Unless the speaker in question allows you to bypass its internal crossover and connect directly to the drivers, you are still using the internal crossover no matter what you do to the signal being fed to the speaker. Removing the "strap" or jumper connection across the bi-wire inputs does not bypass the internal crossover. Furthermore, if a speaker has been designed as an integrated whole including its crossover, an external electronic crossover would have to be designed to mimic the exact behavior of the internal crossover in order to achieve similar performance during bi-amping.
5) I like passive bi-amping a lot, especially the "vertical" arrangement, but I find it usually works best with matched stereo amps and cables.
As with all things audio, Your Mileage May Vary ;-)