oh huni, of course your little one doesn't hate you, he loves you more than anything and you are the one person he relies upon more than anyone else. he likes to be close to you and when you put him down he misses you. wanting a minute to do something for yourself does not make you a bad mother in any way shape or form, your human not a machine!!!
babies are always different around other people then they are with you, its a fact of life, not a sign he doesn't love you. he sees you all day everyday, you are his comfort, his familarity that helps him be who he is. when they see someone else its like wooo hooo more attention.
i used to say the same about my DD, everyone else always seemed to manage to settle her/make her laugh/do things with her better than i could. i swear sometimes she would cry if i wanted to cuddle her.
now (she's 2.4) the bond between us couldn't be any stronger, she's upset or anything and she wants me and i'm perfectly capable of settling her now lol
don't beat yourself up, 6 months is still young and its hard work and stressful at times. maybe your baby is picking up on this a little bit. easier said than done i know but just relax and don't give yourself a hard time, if you do have to put your baby down to do something, put him near you and carry on talking to him or singing to him anything so that he knows you are still around.
big hugs xx