Have you considered that you may be a tad pfb about this 'issue'?
It seems that you heard your mum say 'ah glad you're in a better mood today, I thought you had gone off me and didn't love me any more' on one occasion, and I don't you think need to fear that this remark will linger in your dd's memory or make her believe that people think she dislkes them when she doesn't.
I appreciate that you want the significant adults in your dd's life to be consistent, but that doesn't mean they all have to sing from the same hymnbook all of the time.
Children need to be able to recognise and process differences in the way others react to them otherwise their social skills will be limited and they maybe ill-equipped to handle rejection in later life.
I find it interesting that you've described your dd as 'a bossy and fickle little madam half of the time'. Do you recognise any of her traits in yourself? How do you think you would have reacted if your dm had laid down the law when you were a teenager - and are you being entirely honest with yourself if you say that you would have unquestioningly obeyed the rules or never broken them?
As caricatured in Ab Fab, it's a common phenomena for a lax and permissive parent to produce a strait-laced disapproving offspring. Have you morphed into a bit of a Saffy after your 'horrendous misspent teenage years'? How horrendous were they, I wonder. Did you end up on smack or in Holloway - if so, do tell.
Parents have a duty of care to guide their offspring to adulthood but I don't believe that parents should 'shape' their progeny in the sense that they have the right to 'mould' them into their (the parents) expectations of what they believe their children should be, rather than what the individual child's inherent gifts and talents inclines them to be.
As for not being your dd's 'effing friend', it would be inappropriate for you to burden your child with adult matters until she becomes an adult herself, but you should be her friend in the sense that you are always 'there' for her to support her through all of the trials and tribulations of childhood and beyond.
I'm so sorry that your mum 'had a crap childhood'. It's tragic that so many have never felt wanted or loved by those who brought them into this world and I want to give your mum, and others who've also suffered deep feelings of rejection in their childhoods that have conflicted their adult lives, a very big unmumsnet cuddle.
Why not try giving your mum the reassurance she craves before she asks for it? Never fail to tell her you love her every time you speak to her. Put your arms around her, tell her she's wonderful, organise little treats for her, take her out to lunch 'just because' and above all, relax - you're a great parent and you can afford to take your foot off the brake now and again and let granny do it her way.