I'm curious to know if what I did was right.

I was in a left-turn lane behind 3 cars waiting for the set of left-turn lights (the two lights with the "left turn signal" sign)... It was a much longer wait than usual, and the left turn lights did not turn green after perpendicular traffic had gone through. Instead, the left-turn signal for the traffic on the opposite side of the road turned green for oncoming traffic. For our side, it skipped the left turn signal.

At this point, everybody in the left-turn lane was clearly frustrated. There was no oncoming traffic at all. After a minute, the 3 cars in front of me decided to just turn left while the left-turn signal was red (but there was a green light for traffic to go forward). I did not follow them. A woman behind me was angry and threw her hands in the air and honked at me for not going (and eventually she did an unsafe turn and almost hit my car, for those who are curious). It was safe to turn, but the left-turn signal was red. Should I have gone, or was she just being impatient? Or would I have gotten a ticket if police saw me do it?

Whenever there is signage and/or dedicated turn lane traffic lights, you're required to wait for a green signal to make your turn. In short, you did the right thing by waiting.