The Free Dictionary says:

To do (someone or oneself) a favor means:

To help someone else, typically at their request. In this usage, the person being helped is stated between "do" and "a."

  • Example: Hey, do me a favor and take these bags into the kitchen while I get the others from the car.

while, the Cambridge Dictionary says:

do me/us a favour! as an informal phrase, means:

Something you say in answer to a stupid and impossible suggestion:

  • Example: "Why don't you tell the police what happened?" "Oh, do me a favour!"

These are two absolutely different matters, and I'm looking for a formal (Cambridge introduces this phrase as an informal one), and not something ambiguous (while different dictionaries have different explanations of it).

Is this expression frequently used in "formal situations" in my first example sense?
If not, then I wonder if there is any better formal and NOT ambiguous substitute for that in my first scenario.

  • 2
    Cambridge does list the first sense (B1). It doesn't call it an idiom because it isn't one (all the words used have their ordinary meaning). In that sense could you do me a favour is not particularly formal or informal.
    – user96060
    May 29 '19 at 5:30

The two examples / definitions are not ambiguous and can be both used, in the right contexts.

The definition you found in the Free Dictionary is the generic one.

For the definition in the Cambridge Dictionary, you must notice that it is a different topic actually. There is an exclamation mark at the end!

It means that the idiom is used as a statement - and that statement has the meaning described.

The two meanings are similar, the second one being used with sarcasm.

To mix the statement and the explanation in one sentence, we would get:

Do me a favor and don't be stupid!

With this I try to show how both definitions fit, each of them to the proper context.

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