The list of LW songs in iTunes is simply a playlist. The songs themselves don't exist within iTunes's Music Library which is what you want. So there are various techniques to get the songs into iTunes. By the sounds of it you either didn't understand the techniques described in the above link I gave you or you didn't read thru it. (Oh & just because LW's playlist is in iTunes, don't confuse with thinking the songs exist in iTunes Music Library & then delete the songs. Remember that is simply a playlist which points to the location of the songs which is in your LW downld folder.) You need to add them but see the many techniques from the link above.

You can add them directly from LW's Library window if you want to. Just drag & drop into iTunes library.

The reason LW's media player plays when you play a song from LW is that you have the LW player enabled. If you'd prefer to disable it so your default player (iTunes) plays them instead, then go to your LW menu Tools>Options>Player & untick option box & press apply. However this won't take effect until next time you open LW.