Help General confussion with the 承る family

I'm a little confused with the meaning of these three words: 承認、了承、承る.
After checking some dics and sites on the internet this is what I have:

承認 :
To recognize something that has been justified or it's a fact (like "approved" or "checked and confirmed"). It's also usually used by a company or corporation (possible synonyms: 認める、OKです).

Click to expand...

了承:
To be completely convinced about the circumstances of something. I don't understand this one, but I think It could be something like "there is a plan or a schedule being the object of a discussion and finally, It gets approved".

Click to expand...

承る:
To know something only by rumors or "what my friend told me", but it's not a fact neither something I truly understand.

Click to expand...

bonus:

承知:
To understand or comprehend something very very well. Possible synonyms: よくわかる、知る、心得る、理解.

Click to expand...

I've read tons of example sentences but I am not sure of having understood completely all the meanings.
Thanks in advance~

承認 is mostly to approve like "approve" a bill. There's some sort of formal endorsement involved and suggests there is real power behind the approval or denial.
了承 is broadly to approve, accept, or understand. It is partially synonymous with 承認, but is also used to accept/understand something that is not in the subject's control.
承知 fundamentally means to know or be aware of something, but carries a secondary meaning of "to acknowledge", and can be used to accept commands or requests from others, or even to give permission, or to forgive.
承る is fundamentally a humble term for "to hear" and can be used as a humble replacement for 聞く. By extension, saying "I hear you" in a humble way has a secondary meaning of "(I/we) hear your request/command and will carry it out".

I may be wrong, but I feel like you're studying the kanji and just looked up all the words it is used to spell. I'd really recommend against this, studying several words together that are close in meaning from their definitions with little context is more likely to create confusion than to resolve it. If your dedicated to course of learning kanji, I'd recommend picking just one common, useful word for the kanji to begin with (承知 would be my pick here, although none of these words are very basic). Otherwise, I would say to study words only after you encounter them in a more-or-less natural context (native materials, conversations with natives, or textbook dialogues/graded readers and other such practice materials).

That said, I think these dictionary entries are fairly illustrative of typical uses of the words.

Thanks for the complete explication of all the terms. In fact, I've just looked up for the meaning of 承る on the goo dictionary and I was amazed by how they explained all the possible meanings. Thanks again for the links.

For the other thing you said, well, Indeed I'm studying the vocabulary of the N2 level, so these were the words I have to know, there is no other option.

SomeCallMeChrisの助言を承りました。

どうもありがとうございました。