I will rewrite using Gary's terminology. The TL-POE10R POE Splitter has 3 connections: (1) RJ-45 labeled "power+data in", (2) RJ45 labeled "lan out", and barrel connector labeled "dc out". Power is supplied to the POE splitter via the "power+data in" connector. The "lan out" connector is not POE, i.e. there is no DC power provided. I do not know of any POE splitter that provides power at the "lan out" connection.
The "correct" way to use the POE splitter is to connect the splitter's "lan+data in" to a poe switch, "lan out" to the camera's ethernet connection, and "dc out" to the camera's 12 volt power input. An external
IR light is parallel connected with the "dc out" connection to the camera.
What I did as an experiment was to connect the POE switch output to a 1-to-2 ethernet cable splitter that provides all 8 wires to both of its rj45 outputs. I plugged the camera into one of the outputs, and the POE splitter's "lan+data in" to the other. (Two devices hooked in parallel to the single POE switch port.) The POE splitter's "dc out" powers the external IR light. The POE splitter's "lan out" is not connected, and the camera's 12 volt power input is not connected.
The confusion with the ethernet cable splitter's name is two different types are called by the same name: (1) One type that supplies all 8 data lines at the input to two or more outputs (the type I use in this thread), and (2) The other type that uses the brown and blue pairs of an ethernet wire to carry 100 mb data between a second switch port and a second ethernet device. The second port's orange and green pairs are re-routed to the blue and brown pairs at the switch end of a cable, and "un-routed" at the far end of the cable.