When you clear Data Frame's coordinate system, ArcMap reads any data as it would be provided in Map's units, defined in Data Frame Properties -> General. It will completely ignore data's coordinate systems and there will be no projection on-the-fly. The same data, one in meters and one in US Feet will be offset for a lot and data in geographic coordinates will be very very small, located in null-island. Basically nothing will line up.
When you add more data after clearing a coordinate system, that will overwrite your previous selection and your Data Frame's coordinate system will be the one of your new data. Since Define Projection tool does not add new data, I assume this tool does not overwrite the cleared coordinate system, but Project tool most likely does. I suggest you add all your data you wish to explore before clearing a coordinate system.
As I mentioned above, clearing a coordinate system from your map is risky and you really need to know what you are doing. Tools do not work properly and it might lead to data corruption. Therefore, ArcMap overwrites cleared coordinate system every time it can.