No, you are not lax.
You are wisely choosing to lead by example, give small but unpointed reminders, & to NOT set up a win/lose game. They only lead to battles.
Your relatives are doing it Old School - making D feel that saying "please" is a battle, not an option, & that if she complies she has somehow lost. OK - she won't be reasoning all that out yet, but she will be feeling it.
It's very human & normal to resist something somebody else is badgering us to do, when we don't perceive the point of it.
Your way is bound to lead to better results, it will just take time.
My GD is raised by my DD on a "don't sweat the small stuff" principle. So no food battles, just options put in front of her, & she tries a bit of everything & leaves what she does not want to eat. Because that allows the child feel in control at mealtimes, & helps mitigate against eg food refusal, fussiness, whining, feeling hard done by ...
Another good example of this was at xmas, when GD wanted to do something - can't remember what, not important - her mum said "sure, but we'll put away all the painting stuff first".
That obviously sounded a right bore to GD, so she ignored it & trotted off to another room to start activity X. I went to retrieve her, & my DD whispered "leave it!" - so I backed off. Lo & behold - GD came back in & started putting away her painting stuff, because being alone in the room with activity X was more boring that doing what the grown up asked for ... 
You are using psychology. Your relatives are using - in effect - brute force.
They are capable of physically withholding what your DD wants.
I appreciate that to them it feels like a battle that must be "won" & "it's for her own good" etc - but it really isn't.
Ignore the bad & reward the good. Simple stuff - although not always 100% easy to remember!
TL:DR you are setting up a win/win situation, where DD gradually starts to enjoy saying please & thank you because grown ups reward her for it with smiles & the things she wants. Kids start to really like the 'control' they eventually see as coming from them CHOOSING to say please. It gets them positive attention, AND the things they want!
Your relatives - a win/lose game - where DD loses face, & prefers to sacrifice her 'thing' to maintain her frustrated ego in the face of what seems like unreasonable behaviour from her perspective. No wonder she can only express herself by getting so upset she eventually tantrums, poor little love!
As to how you convey this to relatives ... phew, more info about them needed, & I'm sure wise PP will be along soon to make suggestions.