I might have led you astray by comparing the Paula to the SID. Paula is all about the samples. And I wouldn't call it synthesis really, but playback. As I understand, any actual synthesis on the Amiga would have to be software based, waveforms calculated in a program and the results routed as digital samples to Paula's channels for outputting. That's how I think AHX works. (A little help here, Amiga peeps?)
Then there's chiptunes, which might sound like the music on older, 8-bit systems, but are actually sample based modules using tiny looped sample fragments to create sound wave forms. So, chiptunes and basic synthesis on the Amiga is actually the first wave of C64 nostalgia in music. "Real" sounding digital music came first.
Goattracker integrates a SID emulation engine, so it should be pretty close to the real thing. And sure there's Amiga emulation, in form of (Win)UAE. Just like you can run a C64 music editor in a C64 emulator on the PC, you can do the same with Amiga. But most people don't feel the need to as tracking already moved on to the PC in the 1990s, Amiga emulation requires a lot of CPU power which isn't available on older comps and portable devices and using native apps just generally is a whole lot smoother.
I think you got it already, but there's no difference between custom drawn waveforms and regular samples. Drawing is just another way of creating samples, a feature copied over from Fasttracker II, one of the most popular DOS trackers.
MIDI playback on PC depended for a long time on solely the sound card. For most people this meant the Yamaha OPL FM chip on SoundBlaster and Adlib sound cards which could be compared to the Sega Mega Drive (having another Yamaha FM chip) or the previous generation mobile phone polyphonic ringtones, you know before MP3s got there too. If nothing special was done, the MIDI music would be played with an FM approximation of the General MIDI bank. Some game developers/musicians would go through the trouble of coming up with a proper music drivers for the OPL chip and programming the chip's register to create custom instrument patches and as a result, more inspired FM arrangements of the score. A good example of this is the PC version of the game Dune 2. A lot of PC game music was optimally targeted for MIDI sound modules like the MT-32 and Sound Canvas manufactured by Roland but the cheaper all-in-one consumer solution, SoundBlaster, is what most people had. As the OPL chip was the prominent music source in PCs of the first half of the 1990s, the demoscene got there too. There are plenty of trackers for that synth chip too, but that action is about programming the OPL registers and has nothing to with MIDI.
On the other hand, demoscene people got their taste of MIDI music with Wavetable synthesis when Gravis released their Ultrasound sound card and gave them away to people at demo parties. But the scene people would be more interested in the digital sound mixing that the card was able to do itself without taxing the CPU. So, soon we had demos with digital tracker music played with high sound quality and visual effects running on the screen blazing fast thanks to the sound card having taken the sound mixing load off the CPU.Wavetable synthesis is what you hear on a run-of-the-mill Windows PC today, although the default sound bank in DirectMusic (a Microsoft DirectX component handling MIDI music) is low quality and pretty effin' horrible.
Lastly, I understood you essentially asked how a musician chooses or dictates what their MIDI music sounds like. Well, they don't. As MIDI is only musical information, the sound of the playback always depends on the hardware/software setup.