What is the correct definition for Mindfulness? I was reading a book where author defined Mindfulness 
(sati) using these three analogies from different suttas: 



  The Buddha compares mindfulness to a gatekeeper for a frontier fortress. [Kimsuka Sutta]
   


  There’s another passage where the Buddha says that mindfulness is like a goad. Most of us have gotten away from animal husbandry and farming, and so we don’t even know what a goad is. It’s a long stick with a sharp point. You use it to poke your animals when they’re going the wrong direction, or if they’re standing still when they should be going. The implication here is that the ability to remember what’s skillful and what’s not, and to be able to give yourself a little push or a poke in the right direction: That’s what mindfulness does for you.
  
  And sometimes it’s more than just a little poke.


(May be he is referring to Patoda Sutta. But I'm not very sure because Buddha didn't mentioned Mindfulness in this Sutta. Please provide me correct sutta if there is.)

  There’s another passage where the Buddha says that when you see that something unskillful has arisen in your mind, then you act as if your hair were on fire. You do everything you can, as quickly as you can, to put it out. You’re relentless and mindful in being focused on putting out the fire, and nothing else.



And other definitions which is written in Wikipedia:



  Mindfulness is the psychological process of purposely bringing one's attention to experiences occurring in the present moment without judgment.


(Authour refers it as equanimity) 


But as I understand, I will only call Gate-keeper analogy as Mindfulness.
And other analogies falls in the category of Right Effort and Right Resolve. 

If we look at MN 117 : 


  "One makes an effort for the abandoning of wrong resolve &amp; for entering right resolve: This is one's right effort. One is mindful to abandon wrong resolve &amp; to enter &amp; remain in right resolve: This is one's right mindfulness. Thus these three qualities — right view, right effort, &amp; right mindfulness — run &amp; circle around right resolve." 


Please correct me if I misunderstood this.
Sati

That which is mindfulness, recollection, recall, mindfulness, remembrance, bearing (in mind), not losing, not confusing, mindfulness, the Faculty of Mindfulness, the Strength of Mindfulness, Right Mindfulness: this is called ‘mindfulness.’
Therein what is mindfulness-awakening-factor? Herein a monk is mindful, furnished with excellent mindfulness-penetration, he remembers, remembers constantly, what has long been done and long been said (concerning release). This is called mindfulness-awakening-factor.
Unshakability by negligence as the meaning of the power [strength] of mindfulness
The mindfulness which on that occasion is recollecting, calling back to mind; the mindfulness which is remembering, bearing in mind, the opposite of superficiality and of obliviousness; mindfulness as faculty, mindfulness as power, right mindfulness—this is the faculty of mindfulness that there then is.

These are theravadin abhidhamma and patisambhidamagga excerpts

The Blessed One said, “Suppose, monks, that a large crowd of people comes thronging together, saying, ‘The beauty queen! The beauty queen!’ And suppose that the beauty queen is highly accomplished at singing &amp; dancing, so that an even greater crowd comes thronging, saying, ‘The beauty queen is singing! The beauty queen is dancing!’ Then a man comes along, desiring life &amp; shrinking from death, desiring pleasure &amp; abhorring pain. They say to him, ‘Now look here, mister. You must take this bowl filled to the brim with oil and carry it on your head in between the great crowd &amp; the beauty queen. A man with a raised sword will follow right behind you, and wherever you spill even a drop of oil, right there will he cut off your head.’ Now what do you think, monks: Will that man, not paying attention to the bowl of oil, let himself get distracted outside?”
“No, lord.”
“I have given you this parable to convey a meaning. The meaning is this: The bowl filled to the brim with oil stands for mindfulness immersed in the body. Thus you should train yourselves: ‘We will develop mindfulness immersed in the body. We will pursue it, hand it the reins and take it as a basis, give it a grounding, steady it, consolidate it, and undertake it well.’ That is how you should train yourselves.” https://www.accesstoinsight.org/tipitaka/sn/sn47/sn47.020.than.html