For the most part, the Cambridge definition is correct when talking about things that are literally burning.
Every few years a massive fire breaks out in the open areas around my city. Although they are miles away, the flames grow so large that we can see them clearly from my house.
Figuratively, the two are more or less synonymous. Flame is somewhat more poetic than fire, and may sound better when writing about strong emotions:
He was shy and never spoke to her, but every day a secret flame burned in his heart each time she passed near.
Since it's inelegant to use the same word more than once in a sentence, writers often switch back and forth between the two, albeit sometimes to excess:
"I am a servant of the Secret Fire, wielder of the flame of Anor. You cannot pass. The dark fire will not avail you, flame of Udûn. Go back to the Shadow! You cannot pass." -- J.R.R. Tolkien, "The Fellowship of the Ring"
The point is that if Tolkien had instead written,
I am a servant of the Secret Flame, wielder of the fire of Anor ..."
no one would have noticed any significant difference. It just sounds a little better the other way around.