It has to do with it being an informal way of speaking. There is indeed nothing grammar wise that states the request part, however the meaning is implied and is inferred by the listener due to it not following up with an action following the ないで part. (and context / tone of voice / …)
For examples on dropping the ください, when looking at the entry for ください in A Dictionary of Basic Japanese Grammar, we also find this note :
- In very informal speech, kudasa; may drop. (This form of request is
often used by female speakers.)
(2) a. 早く来て。
Hayaku kite.
(Please come quickly.)
b. まだ帰らないで。
Mada kaeranai de.
(Please don’t go home yet.)
If we’re looking at the second link you provided in the opening text the relevant parts are :
You can remove the ください (kudasai), to just say ないで (naide), which is more casual.
Be careful to not confuse this with the other ないで (naide) meaning.
Let’s look at some examples to make this clear :
To state "Do not"
If we’re working with the “Do not” meaning we will have two clauses, (A) and (B), where (B) goes into more detail about the “without doing (A)” part.
休ま(A)ないで仕事(B)をする。
So we have “to work (B) without taking a break (A).”
(A) ないで (B)
「後悔し(A)ないで生きたい(B)。」
So, we “Want to live (B) without regret (A)”
(A) ないで (B)
To state (please) do not
姉が弟に言う:「頼むから、親には何も言わないで。」
As we can see here, we still have a clause stating “without telling (our parents)”, but we have no clause for more information on the action. We are simply ending on ないで without any action following.
母親が子供に言う:「いい子だから、泣かないでね。」
Again, a clause for “do not” (cry), no second clause with more explanation on the action following.