Short answers, pulled from the story.
The Mahasaghika school declared the Buddha's power to be immeasurable and irresistible. This tradition taught that worship alone suffices for attaining Nirvana without relying on self-effort.
Tanluán lived between 476 and 542 and became the first thinker to apply other-power to Pure Land context. He defined other-power as the dominant causal condition for attaining complete Buddhahood.
Shinran lived from 1173 to 1263 and made other-power central feature of his thought with total rejection of self-power. His worldview requires recognizing one's foolishness and inability to reach Buddhahood through own efforts.
Yúnqī Zhūhóng lived between 1535 and 1615 explained relationship through concept of sympathetic resonance. Yuán Hóngdào lived from 1568 to 1610 used similes like mighty wind producing noise in apertures.
Mantrayana period applied idea of Buddha's power influencing practitioner to tantric practices like mantras mandalas initiation. Root Manual of Rites of Mañjuśrī states power of all buddhas manifests as accomplishment in activities involving mantras.