Meaning and Definition of Paṭisambhidā-magga-pali
pați = bond in order to reflect back
sam = with
bhidha (bheda) = breaking / analyzing
As regards to Ven.Sariputta’s way, it is with
magga = path
pali = utterance / exposition
Thus, Paṭisambhidāmagga would represent as “to reflect upon the meaning of words, it’s nature (dhamma), it’s etymology and fluency to make it understandable” which leads to path and fruition in a practical way.
Reference: Paṭisambhidāpattasutta
Detailed exposition of above sutta:
Pañcahi, bhikkhave, dhammehi samannāgato thero bhikkhu sabrahmacārīnaṁ piyo ca hoti manāpo ca garu ca bhāvanīyo ca. Katamehi pañcahi?
Bhikkhus, a senior monk endowed with five qualities is dear and agreeable to his fellow celibates, respected and esteemed. What are these five qualities?
Atthapaṭisambhidāpatto hoti, dhammapaṭisambhidāpatto hoti, niruttipaṭisambhidāpatto hoti, paṭibhānapaṭisambhidāpatto hoti, yāni tāni sabrahmacārīnaṁ uccāvacāni kiṅkaraṇīyāni tattha dakkho hoti analaso tatrupāyāya vīmaṁsāya samannāgato alaṁ kātuṁ alaṁ saṁvidhātuṁ— imehi kho, bhikkhave, pañcahi dhammehi samannāgato thero bhikkhu sabrahmacārīnaṁ piyo ca hoti manāpo ca garu ca bhāvanīyo cā”ti.
- Atthapaṭisambhidāpatto hoti – He is one who has attained the analytical knowledge of meanings (atthapaṭisambhidā).
- Dhammapaṭisambhidāpatto hoti – He is one who has attained the analytical knowledge of dhamma through language (dhammapaṭisambhidā).
- Niruttipaṭisambhidāpatto hoti – He is one who has attained the analytical knowledge of etymology (niruttipaṭisambhidā).
- Paṭibhānapaṭisambhidāpatto hoti – He is one who has attained the analytical knowledge of eloquence and depth (paṭibhānapaṭisambhidā).
- Yāni tāni sabrahmacārīnaṁ uccāvacāni kiṅkaraṇīyāni tattha dakkho hoti analaso tatrupāyāya vīmaṁsāya samannāgato alaṁ kātuṁ alaṁ saṁvidhātuṁ – He is competent, not lazy, and skilled in tasks both high and low among his fellow celibates, and he possesses the ability to investigate, carry out and organize the above at ease.
Imehi kho, bhikkhave, pañcahi dhammehi samannāgato thero bhikkhu sabrahmacārīnaṁ piyo ca hoti manāpo ca garu ca bhāvanīyo cā”ti.
Endowed with these five qualities, a senior monk is dear and agreeable to his fellow celibates, respected and esteemed.
Let’s start the journey with. Ñāṇakathā 
Style of Ven.Sariputta can be found in Dasuttara Sutta in Dhiga Nikaya. An entire exposition of the same can be accessed here.