i would say simply "i ran home." if what you want to indicate is running using your legs. "rushed" could mean all sorts of things, but might also be good.
Off topic but perhaps not completely since we're discussing clear written expression: "died in an car accident" is the default rendering of such an event in news reports, yet "died in a car crash" is more accurate. What makes a crash an "accident" by default? I have long believed this is powerful manipulation by the car and oil industries to lesson the impact — as it were — of the word "crash."
I think it's more the legal notion of intent that separates manslaughter versus murder. I would venture to say most car crashes that aren't attacks are not intentional, therefore they are accidental. Also if you experience a fatal accident in a car, it's likely enough to be a crash - so I would also venture to say most people understand "car accident" to strongly imply "crash".
I would regularly say: "I ran home as fast as possible", while a native speaker would just use: "I rushed home".
On the second reading, my first sentence may read: "That's one of the examples that give out non-native speakers".