Mary Shogun

Hi there ! Ok, well I'm trying to emulate harmonies from my favorite covers songs so I can record them to sound as close to the original as possible. I have read that harmonizing is merely either a third up or down from the original melody keeping in mind whether the original song is in a minor or major key. So, in my 'wisdom' I thought ok, well I'll just use the transpose button on my keyboard piano, turn that down 4 semitones to get the harmony a 'third down' and I should be right. So I did this, sang along to this melody to a chorus of a very popular song, recorded it and felt it sounded great. NOW, I get to talk to an experienced 'musician' and he says no, that's wrong. I have to only sing the notes that are in the original melody's upper third or upper sixth note and of the same key. Ok, so if this is true, how come one of the songs I do, has the harmony exactly one octave down, singing exactly the original melody? Is there an easy way to work all this out, like via my keyboard as I was happily doing before? Is there a notation program? This harmony thing is becoming quite stressful as I felt it would be relatively easy. Just find the original melody by ear, write down those notes and then transpose them either up or down a third and sing along. Where am I going wrong? I'm quite frustrated and really want to be able to sing good and correct harmonies to my songs as soon as possible. Thanks for any help especially focussing on harmonies LOWER than the original melody, as in the Steely Dan song, 'Rikki Don't Lose That Number' Thank you, any help would be greatly appreciated. Mary