Background: This is my 3rd season riding, but the first where I can say I truly feel comfortable on the bike and gaining confidence in what I'm doing. I'm pretty cautious in all areas of my life, not just riding. I took the MSF course and very much ride within my abilities. I wear ATGATT.
Incident: Today on a group ride in the city, speeds of ~30-35 mph tops, I was the 3rd person in a staggered formation (so effectively behind the leader). An elderly woman apparently did not see all of us and pulled her car out in front of the group very quickly and unexpectedly in a way that her car was horizontal across the road, taking up the entire lane we were riding in (narrow city streets). We were also still accelerating as we had not been on the street very long (I was in 2nd gear). Instead of making conscious decisions about how to brake, I acted on instinct. My only thoughts were, damn I don't want to go down in front of all these other riders, and I certainly don't want to take any of them out with me. What I think I did (hard to remember exactly in these situations) is pull in my clutch (at MSF they drilled in to always pull in clutch if you're hitting the brakes), and brake relatively hard but evenly on both my rear and front brakes. My rear tire skidded, but luckily I kept it upright and all was well, probably thanks to the low speeds. Another rider said my tire smoked. I didn't come to a complete stop but slowed significantly and then once straightened out, maneuvered around the car, and continued on riding fine.
Question: How should I have braked to avoid that skid and retain more control? How can I practice this so instinct next time is to do that instead of what I did today (I have access to large empty parking lots)? Any other general emergency braking advice? I brake just fine in non emergencies.
Thanks in advance!