In my (somewhat limited) experience, the equipment does make a difference.

I used, years ago, to record my singing on a cheap cassette tape recorder - and it was, as you say, awful. More recently - well, 10 or more years ago now, I shudder to realise - I made a demo CD with just 4 tracks on it, using all the proper super duper digital systems (DAT to CD) and felt that the product was much more like my real voice. A month ot two ago I recorded one song on a CD using the most up to date computer-based system, as was astonished to hear coming back at me something that sounded exactly like what I hear when I sing.

So - don't despair: maybe your friends are telling the truth!   The way to tell, I reckon, is to listen to the intensity and duration of the applause when you've finished.

Steve