Does accusation helps a person change his values or a better ways is to tell a story with moral?

"Kesi, I train a tamable person [sometimes] with gentleness, [sometimes] with harshness, [sometimes] with both gentleness & harshness.

"In using gentleness, [I teach:] 'Such is good bodily conduct. Such is the result of good bodily conduct. Such is good verbal conduct. Such is the result of good verbal conduct. Such is good mental conduct. Such is the result of good mental conduct. Such are the devas. Such are human beings.'

"In using harshness, [I teach:] 'Such is bodily misconduct. Such is the result of bodily misconduct. Such is verbal misconduct. Such is the result of verbal misconduct. Such is mental misconduct. Such is the result of mental misconduct. Such is hell. Such is the animal womb. Such the realm of the hungry shades.'
AN 4.111

I've heard that if you have something to run from and something to run towards, then you will run faster having both kind of motivations compared to either avoidance or reward promted running.

What you want to avoid is your words being perceived as a punishment of what you think is bad behavior.

You really want to avoid this because you will most certainly be met with retaliation and it won't work properly. At best you may have enough leverage to force restraint but this is compulsion based training and isn't proper.

As Oya Mist touched upon, you should be extremely careful when dealing with people. That especially if you are trying to do good things, really... bad people will straight kill you... no really, they will and i think most people don't realize how easy it is to trigger people to want your life.

First, I beware of "A or B?" questions -- because that could be a 'false dichotomy', and a better answer might be "both", or "neither", or "something else entirely" -- or perhaps the question is unanswerable in its present form e.g. without more information.

Second, I don't see how "telling a story with a moral" is supposed to work: because either they don't recognise their own behaviour in the story (in which case it's ineffective and a waste of time); or they do recognise that the story is intended as a critique of their own behaviour, in which case how is it any different from an "accusation"?

Third, when we read the suttas we are reading stories about other people. But the dialogs in the suttas tend to be people talking to each other, they're not telling stories or talking about other people. I'm saying that "telling stories and talking about other people" doesn't seem to be the way that people talked to each other in the suttas.

Some of the teaching in the suttas takes the form of asking questions.

Perhaps "asking questions" might be one of the ways to appear less offensive -- instead of imposing your view on someone (e.g. "You are flattering this person"), you are asking for their view, like asking their advice ("Are you flattering this person?"). So perhaps a line of questioning like, "Are you flattering this person? Is that a good idea? Are you worried that might have such-and-such disadvantage?" is a way to have a conversation with the person about your concerns, while being less of an accusation, and more respectful in that it's asking them to share their view instead of imposing your own view.

And if they do answer, maybe avoid using their answer as an excuse to argue with them!

Another way to try to avoid offence is to use "I messages" i.e. talk about "I" -- for example not, "You're wrong to flatter people" (which is a "you" message), not even "Flattery is wrong" (which is still an accusation), but "I am afraid that flattery has such-and-such disadvantage" (which is an "I" message). And if you're going to tell a story with a moral, then a story based on your own personal experience might be seen as friendlier (and perhaps more truthful) -- "sharing your own experience", for your welfare and for theirs, instead of "moralizing".

See also What is the Buddhist view in Socratic questioning? on the subject of "asking questions" (and of "teaching by asking questions").

According to AN 4.42 there are a variety of ways to answer questions, and I expect it's done more or less skilfully. Note also AN 5.159 which says teaching should be with the thought of speaking "without exalting oneself or putting other people down", and isn't easy. The whole topic might be linked to "conceit" -- see Māna (Wikipedia)" and How are 'conceit' and 'identity-view' not the same? -- and difficult to perform well.

I don't think there's a "one fits all" answer to this. That's why in Buddhism the teacher can avail him/herself of many different upaya. A teaching method that will fit one student doesn't necessarily fit the other. If that was the case, everyone would gain instant attainment after hearing an enlightenment story like a Zen Kōan.

This is why the personal relation between teacher and student is so important. It is likely also part of why there are so many different branches of Buddhism.

The Buddha indicates that it is inappropriate to talk in certain ways to certain people in certain conditions:

AN5.157:1.1: “Mendicants, it is inappropriate to speak to five kinds of person by comparing that person with someone else.
AN5.157:1.3: It’s inappropriate to talk to an unfaithful person about faith.
AN5.157:1.4: It’s inappropriate to talk to an unethical person about ethics.
AN5.157:1.5: It’s inappropriate to talk to an unlearned person about learning.
AN5.157:1.6: It’s inappropriate to talk to a stingy person about generosity.
AN5.157:1.7: It’s inappropriate to talk to a witless person about wisdom.

If your friend gets upset, then your intervention would be inappropriate. For example:

AN5.157:6.1: And why is it inappropriate to talk to a witless person about wisdom?
AN5.157:6.2: When a witless person is spoken to about wisdom they lose their temper, becoming annoyed, hostile, and hard-hearted, and displaying annoyance, hate, and bitterness.

So be careful to talk with the right person at the right time about the right things.

As already told by householder 'letsbuddhism' there are three ways of teaching: telling benefits, telling drawbacks, and a combination. If either wouldn't help, possible no way that taking on.

As for the quote from SuttaCentral of AN 5.157 in another answer: first, the translation that talking is "inappropriate" is better translated as "burdensome" or "troublesome"; second, someone who is unrelated has no duty to teach, someone who is a relative does have.

This is told although knowing that the 'Robin Hoods' lack of wisdom and take it as food for their anger, yet there is still a small opening, that one could pay right attention.

Sure, if in dependency one, if not able to let go, should take care not to lose it, even if well meant.

So all three kinds of teaching should be traceable here as an account, yet just for wise and those in relation toward the Gems, or after it.