You shouldn't have took her to car if you weren't prepared with condom. Very low game plan mate. You made her feel insecure when you pulled out and even enforced her logic about men. So when you pulled her away, no matter if it was due to no condom, this created resentment which is why the awkward silence. You killed the comfort you created right down to zero.
That's why she wanted nothing to do with you afterwards but you persisted, in a needy, clingy way and tried to even talk about a threesome, which enforced the logic in her mind. You tried to change her mind not...her mood. The mood was dead after that pull back.
It's not just about backing off. You need to understand why it happens or you'll continue doing the mistake.