HI, Mr.
I did come across both sites with their different answers. I'm not going to give an answer as I have had a problem using the verb 'promise' as it has different uses as is mentioned in 'Longman' and 'Advanced Grammar in Use'. I mean, should it be followed by an indirect object + to, infinitive or not? However, I have found these patterns:
1) promise + someone + something:
Ex:
a) My parents promised me a new car. (Longman)
b) The company promised us a bonus. (Cambridge)
2) promise to do something:
Ex: Ann: I will pay you back next week. Really I will. = Annpromised to pay me back the following week. (A Practical English Grammar)
3) promise (somebody) that .....
Ex:
a) Promise me (that) you won't tell him. (Cambridge)
b) The government have promised that they'll reduce taxes.
4) promise someone to do something: (Not OK according to Longman):
Ex:
My parents promised me to buy a new car. (wrong)
My parents promised me that they would buy me a new car. (correct)
So, according to all mentioned, we can say:
a) She promised to give me back my money. (There's a similar example in Collins English Dictionary.)
b) She promised (that) she would give me back my money.
However, I do doubt 'She promised me to give the money back.'
(A Practical English Grammar):

Advanced Grammar in Use considers 'Sam promised me to ....' wrong
