This topic invariably triggers lively debate!
Some say you can simply connect from the headphone socket of a cassette player to the Line In socket of a PC and use the Sound Recorder.
Others (me included) say that doesn't (and indeed cannot) work because the sound level is far, far too low for re-recording purposes.
Use the microphone socket instead and, as you say, that invariably results in massive overload and mega-distortion.
Have you any way of finding out what the "Grey socket with an arrow thingy" (!) is for? If it's an OUT socket, then clearly it can't help. The LINE IN socket is normally black.
I'm afraid I don't have a definitive answer for any of this except to say I've never got it to work satisfactorily either. But depending on whether you wish to throw serious money at it, I see Maplin have (potentially) the very thing, with a USB connection to the PC, and just reduced to £99.99, see click here HTH.