Sorry realised I hadn’t answered your questions. I don’t have a different definition of what a synthesizer is but I think a synthesizer should be a what synthesizers started out as. Sounds based on sin/square/saw etc. Modern synths using other forms of synthesis should be appended with what form of synthesis they use. The microgranny refers to itself as a granular synth for example and so I know what I’m getting with that.
Sorry to disagree but I don’t agree that samplers are synthesizers either, some samplers can be synthesizers like Akai rackmounted samplers for eg but I don’t think I ever transposed a sample across a keyboard on an sp404 ( be interesting if I could let me know if you can)