All of us have different problems in our relationships with our parents.
I am not going to even begin to explain mine with my mom, that would be a novel.
Your mom you said is taking care of your child every other day for 2-3 hrs., have you told her, 'Thanx mom I so appreciate your help'? Sometimes we forget that our parents don't OWE us once we become adults, watching your child is different than spending quality time on her terms with her granchild.
Your friend's mom isn't your mom & there is no point in even going there.
If you realize that you are hurt by your mom's behavior & raise your child without the critisisim that you live with from her then there will be no reason for your child to feel as you do now.
I am 59 yrs young & am raising since 2000 three of my 11 grans now aged 12, 13 & 17 when I got them I had two in diapers, I didn't have 9 months to plan, one day my DH & I were fancy free GP's spending time with grankidz & grown kidz on OUR terms & then...POOF!...instant family, it's been difficult & continues to be so, financially & physically & I, as well have moments when I would almost die for a little help, but my family isn't near me, my DH's family is busy with their lives & so I just do my best & move forward, no time for feeling unloved or unappreciated I am just too busy for that...pick yourself up, dust yourself off, love your child, absorb all the love you can from your child, accept the help that your mother gives in her own way & be grateful that she is there, someday she won't be & you will be remembering all of the good she did not the bad. Make peace with YOU & then find peace with your mother.
Blessed Be...