I know that when giving something, you can say 父に傘を貸してあげました - there's only one word for when giving. But when receiving something, it's either 〜もらいました or 〜くれました.
It's my understanding that, when the subject is myself, we use 〜もらいました, and when the subject is the other person, it's 〜くれました. Is this correct, and are there other considerations that I should be aware of?
Your recognition is totally right. Here are some examples and differences.
私{わたし}は父{ちち}から傘{かさ}をもらいました。 (I got an umbrella from my father.)
You can see that the subject is 私 (I) and もらいました can be applied for this case.
This sentence can imply that the person 私 asked his father to give an umbrella for him/her in several cases.
父{ちち}は私{わたし}に傘{かさ}をくれました。 (My father gave me an umbrella.)
The subject is 父{ちち} and くれました should be appropriate for this case.
Both sentences can imply that either 私 asked him to give umbrella or the father gave him/her an umbrella voluntarily.
In my opinion the second sentence can be associated with voluntariness more than first one.
