Glad to see you got it fixed. Will you be in BA for the Olympics?
I haven't experienced this with the S10 (mine is a 2014 ES too). However, with previous Suzukis, if I recall correctly, certain components in the fuel/exhaust system can be allowed to fail whilst still leaving the bike with normal power output. Some can fail & permit a limp home on reduced power. Then, finally, some can fail & it's game over.
The intake air pressure (IAP) sensor & the Atmospheric (AP) sensor were both in the first category, so the ECU would assume a pre-programmed value if the connector was unplugged, throw the warning light & off you could go to the analyst/dealer for a check up/replacement.
I imagine the S10 will be much the same, but I am a little suprised she didn't throw a warning light/code if not seeing the correct value ranges being fed back to the ECU. When others speak of spitting back/popping it always leaves me to wonder whether they are running the OE headers & pipe, or an aftermarket set up, with/without an Oxygen Sensor on a closed loop, which may not be set up correctly. Equally, an air leak to the inlet or headers can produce similar symptoms.
After all the hard start debate, given the OPs milege/Km, I am assuming there are no other strange issues at start-up.
Ride Safe .................. KEN