Planar heterojunction-based organic light-emitting diodes (PHJ OLEDs) usually have simultaneous exciton and exciplex emissions and understanding their evolution processes is crucial for designing high-performance white OLEDs, but relevant evolution mechanisms are still unclear. Here, unreported competitions between exciton and exciplex emissions and an undiscovered energy-transfer channel from triplet exciton to triplet exciplex states in PHJ OLEDs with different hole-injection abilities are investigated by separately measuring their current-dependent electroluminescence (EL) spectra and magneto-EL (MEL) traces at different temperatures. Interestingly, the ratio of emission intensity between exciton and exciplex states ($R={I}_{\mathrm{exciton}}/{I}_{\mathrm{exciplex}}$) from the device with a good hole-injection ability rises with increasing bias current at each temperature, whereas that from the device with a poor hole-injection ability hardly changes. In addition, R from the devices with good and poor hole-injection abilities show nonmonotonic variation and monotonic reduction with decreasing operational temperature at each bias current, respectively. These various current- and temperature-dependent competitions between exciton and exciplex emissions are attributed to distinct current- and temperature-dependent electron-tunneling effects occurring at organic heterojunction interfaces of devices with different hole-injection abilities. More intriguingly, using MEL as a fingerprint detection tool, we find there is a Dexter energy-transfer (DET) channel from the triplet exciton to triplet exciplex (${T}_{1}\ensuremath{\rightarrow}{\mathrm{EX}}_{3}$) state and the DET can enhance the reverse intersystem crossing (RISC) process from triplet to singlet exciplexes (${\mathrm{EX}}_{3}\ensuremath{\rightarrow}{\mathrm{EX}}_{1}$), which cannot be found using the conventional probing tool, EL spectra. Because the numbers of ${T}_{1}$ excitons in these devices have different current and temperature dependencies, various current- and temperature-dependent DET and RISC processes happen, which cause abundant MEL behaviors. Obviously, this work deepens the physical understanding of the competition and DET processes between exciton and exciplex states in PHJ OLEDs.