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The mechanism of unconventional topological superconductivity (TSC) remains a long-standing issue. We investigate the quantum phase diagram of the extended t-J-J_{χ} model including spin chiral interactions on triangular lattice based on state-of-the-art density matrix renormalization group simulations. We identify distinct classes of superconducting phases characterized by nonzero topological Chern numbers C=1 and 2 and a nematic d-wave superconducting phase with a zero Chern number. The TSC states are shown to emerge from doping either a magnetic insulator or chiral spin liquid, which opens new opportunities for experimental discovery. In addition, we further classify the C=2 class of TSC phases into an isotropic and a nematic TSC phase and present evidence of continuous quantum phase transitions from the nematic TSC phase to both isotropic TSC and nematic d-wave phases. These results provide new insight into the mechanism of TSC with an emphasis on the role played by hole dynamics, which changes spin background and drives a topological phase transition at a hole doping level around 3% upon doping a magnetic insulator to enable the emergence of TSC.