Question
What exactly is dehi according to advaita and vaishnava school?
Answer
Bhagavad Gita is regarded high from Advaita &/or Vaishnava school of thoughts. This answer is from that perspective.
According to it, Dehi (embodied one) transgresses from one body to another, upon death:
वासांसि जीर्णानि यथा विहाय, नवानि गृह्णाति नरोऽपराणि। तथा शरीराणि विहाय जीर्णा, न्यन्यानि संयाति नवानि देही।।
BG 2.22 - As after rejecting worn out clothes a man takes up other new ones, likewise after rejecting worn out bodies the 'embodied one' unites with other new ones.
Almost same description is given for Ishvara as well:
शरीरं यदवाप्नोति यच्चाप्युत्क्रामतीश्वरः। गृहीत्वैतानि संयाति वायुर्गन्धानिवाशयात्।।
BG 15.8 -- When the Body is received and also when given up, the "Master" (ishvara) carries all these [6 senses]; Same as wind transporting smell.
From these 2 verse, we can infer that the Dehi is the consciousness/intellect(Buddhi) or Purusha, which controls the senses + mind like a master.
It's higher in order than the mind, but lower than the Atman (or ParamAtma):
इन्द्रियाणि पराण्याहुरिन्द्रियेभ्यः परं मनः। मनसस्तु परा बुद्धिर्यो बुद्धेः परतस्तु सः ।।
BG 3.42 — Senses are said to be beyond [physical body]; mind (mann) is beyond senses; intellect (buddhi) is even beyond Mind; "That" (Atma) is even beyond intellect.
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