The phase splitter has one input and two outputs. The two outputs are inverted with respect to each other. That is, as one increases in a positive direction the other increases in a negative direction.
In the first diagram the phase splitting is done by the transformer.
In the second diagram, the output from the collector is an inverted version of the input.
The output from the emitter follows the input. There is no amplification from this circuit because the emitter is undecoupled.
The word PHASE indicates a shift in time. In actual fact there is no phase
change here, only inversion of the signal.