Your first script should work. In the second you are missing the preset name and are using the incorrect function export_preset_to (_to doesn't exist) and the third way is not supported at all.
Here is a full example call:
tell application "BetterTouchTool" to export_preset "MyPreset" outputPath "/Users/xxx/Downloads/test.bttpreset" comment "NoComment" link "https://NoLink" with includeSettings
Your script works, but it saves a file equal to 9kb, and my two presents are 50 and 355 kb each, so it does not seem to save either of these. I don't want to import it as it threatens to overwrite my global preferences. I tried to rename "MyPreset" to "Default Main" and "btt-mobile-example" - names of my current presets and it didn't help. What am I doing wrong? I wish it would be a preset file in the BTT directory I can point to - in order to back it up.
I recently realized that backing up the presets isn't enough. In the worst case scenario, I would lose too much. So I back up the BTT folder in App Support. Then you really have everything... hopefully