Then you need to increase the TV device "Power On" delay (to prevent the TV from missing the input command). The reason why you don't have the problem when restarting from the same input, is because it does not send an input command in these cases (Harmony tracks the power and input status of all your devices).

Note: Based on the behaviour of the Help function, it sounds like your TV is configured to use a cycling method (like method 1), instead of the faster and more reliable discrete method (method 2). The Harmony dataset for your LG TV (50PG60UD) has many discrete input commands like InputTuner, InputVideo1-2, InputComponent1-2, InputRGB, InputHdm1-4. Test these discrete commands in device mode paying attention to their behaviour. If they work properly, then reconfigure the TV device Adjust inputs (under Device > Settings > Adjust inputs) to use method 2 (discrete method), which is the Harmony default method for this model (ignore the comment on what the original remote can or can't do. The capability of the original remote is irrelevant. It's the capability of the TV that counts).