What is the definition of 'fool' in Theravada Buddhist text?

From Dīgha Nikāya 16 - Mahāparinibbānasutta The Discourse about the Great Emancipation

The Gracious One said this, and after saying this, the Fortunate One, the Teacher, said something more:

“Youths and also the old, fools and also the wise,
Rich and also the poor—all end in death.
Like an earthen vessel made by a potter
Small and great, that which is baked and unbaked,
All end in breakage, just so life ends in death.”

Then the Teacher, said something more:

“Well-matured, decayed, with little of my life remaining,
Having abandoned rebirth I will go, having made myself a refuge.
Be heedful, mindful, and virtuous, monks,
With well-reasoned thoughts, protect your minds.

Whoever in this Teaching and Discipline will live heedful,
Having given up the round of rebirths, he will put an end to suffering.”

Reading the Sutta, 'fool' is described in many different situations and conditions. Sutta doesn't define exactly what is the meaning of 'fool'. It's us with our own understanding, try to understand Dhamma. Our understanding itself is subject to change.

Endowed with misconducts, someone is a fool. A person who is attached to worldly pleasure also a fool. Even those who have left home for the life of homelessness but is attached to jhana, deviate from true goal, is also considered as a fool, etc..

From my understanding, Buddha encourages us to strive to Further Shore. Be diligent and learn from those who are endowed with purity in conducts. Be compassionate to those who need our guidance as we would want to be treated.

Our practice in helping others should not affect and deviate us from the path, if we are not ready, it's good to take a step back. While learning and practicing, we need to keep wise companion and strive to the state 'devoid of foolishness'.

Edit: Just to add, the word 'fool' is translated into English as 'young child' or childish. It's figurative language to describe someone who is spiritually immature and certainly is not the same as 'fool' in English word.

how to differentiate between fool's misconduct and wise's misconduct?

As per my penetration of the property of Sat-Dharma: fool is someone who thinks that he knows, while in reality not knowing, and acts accordingly. Wise knows what he knows and what he doesn't know - and acts accordingly:

King Milinda: Venerable Nagasena, for whom is the greater demerit, one who performs misconduct knowingly, or one who performs misconduct unknowingly?

Nagasena: What do you think, your majesty, who would get burned more, one who knowingly picks up a hot iron ball, ablaze and glowing, or one who not knowingly picks it up?

Also, Dhammapada 5.69:

So long as misconduct has yet to ripen, the fool mistakes it for honey. But [only] when that misconduct ripens, the fool comes to grief.

-- the implied contrast here is that the wise recognizes his own misconduct as misconduct before it ripens and "drops the iron ball" before it burns.

My commentary: the wise may not know, and can make mistakes - but he understands the limits of his knowledge and acts cautiously, in little steps, verifying his theory from received feedback and adjusting his model accordingly. The fool is assured of his (incorrect) knowledge and is attached to it. He does not actively look out for feedback and does not adjust his understanding and action. The fool is attached to his incorrect understanding.

This is how the fool is setup for suffering: because of his attachment to incorrect understanding which mismatches the reality and provides an incorrect basis for decision-making. The wise adjusts his model all the time to better match the reality, even as reality changes:

Imagine two Roombas (robotic vacuum cleaners), both with incorrect map of the house. One Roomba updates its internal map as it goes around and bumps into things. Another Roomba keeps going based on a wrong map. Which Roomba will clean the house better and which will likely get stuck?

Here is a dictionary definition: Bāla

Bāla1 (adj.) [cp. Sk. bāla (rarely Vedic, more freq. in Ep. & Class. Sk.); its orig. meaning is "young, unable to speak," cp. Lat. infans, hence "like a child, childish; infantile"] 1. ignorant (often with ref. to ignorance in a moral sense, of the common people, the puthujjana), foolish (as contrasted with paṇḍita cp. the Bālapaṇḍita -- sutta M iii.163 sq.; D ii.305 sq.; Vism 499, and contrasts at Sn 578; Dh 63, 64; Pv iv.332; Dhs 1300), lacking in reason, devoid of the power to think & act right.

For what it's worth I learned that a preschool teacher isn't supposed to think or say, "That is a good child" and "That is a bad child." Instead if a child behaves well, "That is good behaviour" and if the child behaves badly, "That is bad behaviour". Having an attitude like that, i.e. that it's behaviour ("misconduct is foolish"), instead of the person ("that person is or is not a fool"), means you don't have to try to discern some artificial distinction between "misconduct of a fool" versus "misconduct of a non-fool".

In SN 12.19 (Bala-pandita Sutta: The Fool & the Wise Person) the Buddha explains "what distinguishing factor is there between the wise person & the fool":

The Blessed One said, "The ignorance with which the fool is obstructed, the craving with which he is conjoined, through which this body results: that ignorance has not been abandoned by the fool; that craving has not been destroyed. Why is that? The fool has not practiced the holy life for the right ending of stress. Therefore, at the break-up of the body, he is headed for a [new] body. Headed for a body, he is not entirely freed from birth, aging, death, sorrow, lamentation, pain, distress, & despair. I tell you, he is not entirely freed from stress & suffering.

"The ignorance with which the wise person is obstructed, the craving with which he is conjoined, through which this body results: that ignorance has been abandoned by the wise person; that craving has been destroyed. Why is that? The wise person has practiced the holy life for the right ending of stress. Therefore, at the break-up of the body, he is not headed for a [new] body. Not headed for a body, he is entirely freed from birth, aging, death, sorrow, lamentation, pain, distress, & despair. He is, I tell you, entirely freed from stress & suffering."