I'll try, but it's kind of confusing, especially since there's no definitive explanation that I've ever seen-- which doesn't mean there's no definition out there, but I haven't seen one-- so it seems to be up to the readers to try to figure out what's meant when a keyboard manufacturer uses the term "part" in their manuals.
In the broadest sense, "part" in this type of usage generally refers to some specific role that a person or object fills in some production-- the part that a given actor plays in a movie, for instance.
In music, the term is used to refer to the portion of a musical work that a particular instrument, singer, or performer plays-- the drums' part, the bass guitar's part, the lead guitar's part, the rhythm guitar's part, the tenor's part, the soprano's part, the first violinist's part, the second violinist's part, etc. When speaking of instruments that are played with both hands, such as a piano, it can also refer to the specific portions of the instrument's part which are to be played by each hand-- the left-hand part and the right-hand part. And for instruments having more than one surface which the performer can play, such as an organ with multiple manuals and perhaps even a pedalboard, or a drum kit that contains a bass drum, two or more snare drums, two or more cymbals, etc., it might refer to the portion of music that's to be played on or with a specific portion of the instrument or kit.
In the case of polyphonic multitimbral electronic keyboards-- that is, keyboards that can make many different sounds or timbres, and can play several different timbres at the same time-- the term "part" seems to be similar to "voice," but not "voice" in the sense that Yamaha likes to use that term, which would more generally be called a "timbre," and which in the case of a synthesizer is commonly called a "patch"; that is, a specific "instrument sound." Rather, "voice" in the sense I'm refering to here really means how many independent sounds can be played at the same time, such as how many voices a synthesizer has, or how many singers (voices) are in a choir. But "part" and "voice" are different.
In an analog synth, each individual sound is generated by an oscillator as a soundwave, which might be combined with or altered by the signals from additional oscillators, as well as shaped by filters, etc. In an electronic keyboard, each individual sound is generated by a tone generator rather than an oscillator, with the tone generator usually able to play a short recording of a sound rather than a continuous soundwave, although the recording can be looped so it seems to continue for as long as the key is pressed (such as an organ sound), or the duration of the recording can be modified by compressing or stretching certain portions of the recording, etc.
A given electronic keyboard has a specific number of tone generators inside, determining its maximum polyphony. But a given instrument sound or "timbre" ("voice" in Yamaha-speak, "tone" in Casio-speak, or "patch" in synth-speak) is often produced using two or more tone generators, so in practice the actual polyphony at any given time depends on how many tone generators are needed to produce each of the timbres that are being played at that time.
Remember, "voice" and "part" aren't the same things, although they're related, so all this talk of tone generators and polyphony is really about voices, not parts.
However, a given electronic keyboard usually has a limited number of different sounds that it will let you play at the same time, and these are its "parts." Each "part" can be assigned a specific instrument sound (timbre, voice, tone, or patch), and can usually make use of as many tone generators as it needs to play all the notes it's called on to play.
In the case of certain digital synths, such as Yamaha's MX line, there is really only one type of part, and there are usually 16 of them, comparable to (but different than) the 16 standard MIDI channels. But in the case of arranger keyboards, there are usually three different types of parts-- the "keyboard parts," the "style parts," and the "song parts." Grouping the parts into these categories is largely a matter of convenience, so all the parts in a given category can be controlled as a group, such as when deciding which parts you'd like to transmit via the MIDI Out channels, or which parts you'd like to control via the MIDI In channels.
In any case, the "keyboard parts" are those parts that the performer can play "live" with the keyboard. Different makes and models of keyboards might use different terms for these parts. For instance, if you play a Genos, Tyros, PSR-SX, or PSR-S model, then you're probably used to hearing these called "R1" or "Right1," "R2" or "Right2," "L" or "Left," and possibly also "R3" or "Right3." On the other hand, PSR-E owners are used to hearing these called "Main," "Dual," and "Split." But other brands might have "R1," "R2," "L1," and "L2"; or they might have a given number of "zones" that can be freely and independently assigned to different ranges of keys on the keyboard.
The "style parts," or more generally the "auto-accompaniment parts" (since some models have additional types of auto accompaniments besides styles, such as "patterns" or "grooves") are the specific parts which are used to play the different sounds of the auto accompaniment. On a Yamaha arranger there are normally eight such parts, which are generally called "Rhythm1," "Rhythm2," "Bass," "Chord1," "Chord2," "Pad," "Phrase1," and "Phrase2," or similar terms ("Drums1" and "Drums2," or "Rhythm" and "Sub-rhythm," etc.).
The "song parts" are those parts which are used for playing back a MIDI song recording, but they're also commonly used to play MIDI data that's being transmitted to the keyboard. Since a standard MIDI file can contain data for 16 channels, and a MIDI port can transmit or receive data for 16 channels, there are normally 16 "song parts," or one for each MIDI channel.
The important thing to remember is that these various "parts" are inside the keyboard, and are basically circuits or computer routines or whatever which are dedicated to generating and controlling the sounds that the keyboard can produce-- not the specific sounds per se, but the musical role that those sounds are filling.
"Channels," on the other hand, are somewhat akin to the lanes of a multi-lane highway, in the sense that they're used for transmitting data and keeping everything separate, sort of like cars and trucks that stick to their assigned lanes. But the channels don't really have their own lanes, like wires that carry specific signals. Rather, channels are more like IDs stamped on cars or trucks or taxis, since the channel numbers are encoded within the MIDI channel messages and are interspersed as they're transmitted, like a single lane (or MIDI port) that might have a Yellow Cab, followed by a Checkered Cab, followed by a Ford Explorer, followed by a different Yellow Cab, etc. So you might have a given MIDI port that sends or receives a channel 1 message, followed by a channel 5 message, then another channel 1 message, then a channel 3 message, etc., depending on which channel needs to play a note at any given time. This isn't a perfect analogy, but the important point is that the channels are being used to send or receive data between the keyboard and some other MIDI-capable device, or between a MIDI song file and the keyboard.
The various parts can be assigned to various MIDI channels, but they aren't the same things as each other. For one thing, on many keyboards you don't even need to assign any of the parts to any of the MIDI channels if you don't want to; you can have it so the keyboard doesn't output any MIDI data, and you might also be able to have it so the keyboard simply ignores any MIDI data that it receives. But if you do want to, you can typically choose to transmit certain parts on the MIDI channels but not other parts, such as sending the keyboard parts but not the style parts, or vice versa; you might be able to choose which parts you want to respond to the MIDI channels; and typically you can decide which part you want to associate with which MIDI Out channel or which MIDI In channel.
I apologize for the rambling discourse, and I hope it wasn't too overwhelming and confusing; but that's basically my understanding of what a "part" is and how it differs from a "channel."
