I'm not an electronic music expert, but I'm confused about synthesis, and how it differs from synthesizer. Is synthesis the process of creating new sounds for a synthesizer? Or something else entirely?
Synthesis is almost always used to refer the the technology inside the synthesizer that actually produces the sounds. Examples would include Analog Synthesis (like the old Roland 303, 202, Juno 6 Synths), FM synthesis (the Yamaha DX7), Sample and synthesis (known under various proprietary names) is the most common, it starts with a sampled sound which is then modified by the internal filters.
The sample methods allow for more simultaneous notes to be played. This is called polyphony. More powerful hardware has led to analog modeling synthesis, which attempts to achieve the widely preferred sound of the old analog boards, without as much limitation in polyphony.
When you are creating the sounds your synthesizer will produce, you are programming the synth. Think of synthesis as the engine of your synthesizer.