What is the correct definition for Mindfulness?
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