Odtnhj

of some idioms, I like: blown out of proportion. TFD

Exaggerated or magnified beyond the true scale or truth of the matter.

As in:

In soccer, players often blow out of proportion their injuries to make their opponent look more guilty.

In the case of the police in particular, as you referenced, to embellish comes to mind: Vocabulary.com

That's what can happen when you embellish by adding too many false or
exaggerated details to a story.

As in:

"Whoa whoa ... whoa! Calm down, sir." after the person with whom they
are speaking says something innocent.

Here the officer is embellishing his response and his authority to the 'words' of a suspect/prep/innocent person.

melodramatising

Meaning 3 seems to fit.

melodrama (ˈmɛləˌdrɑːmə)
n

1. (Film) a play, film, etc, characterized by extravagant action and emotion


2. (Theatre) (formerly) a romantic drama characterized by sensational incident, music, and song


3. overdramatic emotion or behaviour


4. (Theatre) a poem or part of a play or opera spoken to a musical accompaniment

I think you can use ther term overreaction:

Overreact:

to react in an extreme, especially an angry or frightened, way:

• Try not to overreact to criticism.

(Cambridge Dictionary)

As an idiom, those people are making a mountain out of a molehill.

From Wikipedia:

Making a mountain out of a molehill is an idiom referring to over-reactive, histrionic behaviour where a person makes too much of a minor issue. It seems to have come into existence in the 16th century.

Metaphor

The idiom is a metaphor for the common behaviour of responding disproportionately to something - usually an adverse circumstance. One who "makes a mountain out of a molehill" is said to be greatly exaggerating the severity of the situation. In cognitive psychology, this form of distortion is called magnification or overreacting. The phrase itself is so common that a study by psychologists found that with respect to familiarity and image value, it ranks high among the 203 common sayings they tested.

Similar idioms include 'Much ado about nothing' and 'Making a song and dance about nothing'.

I've often seen this described as "hamming up an injury", which is to say they are overacting, rather than overreacting. The distinction there captures the element of deceit that I think you're going for.

A common phrase for this (used by Tar Heels...Blue Devils, Demon Deacons, the Wolfpack, etc.):

cry foul

Of course, this phrase is used by others (according to dictionaries) to mean, for example, this:

Protest strongly about a real or imagined wrong or injustice.

https://en.oxforddictionaries.com/definition/us/cry_foul

Hmm, that's interesting, but in Tar Heel country (US, SE Region, NC), cry foul is commonly used in this context:

People cry foul when they are NOT actually fouled (meaning wronged in some way), often after an unsuccessful attempt to draw a foul and specifically in order to convince others (who did not witness the contact, in person or on video... On video!) that they were indeed fouled, on or off the court, ideally courtside.

BTW, crying wolf is another matter...all together.

I think you've already found exactly the word you need:

Exaggeration

a. The action of exaggerating or magnifying unduly in words or representation.

In soccer specifically, I've seen this called diving or "taking a dive." From Wikipedia:

In [soccer], diving is an attempt by a player to gain an unfair advantage by falling to the ground and possibly feigning an injury, to give the impression that a foul has been committed. Dives are often used to exaggerate the amount of contact present in a challenge.

Could also be simulating

to make a pretense of; feign:

Or drama queen.

a person who habitually responds to situations in a melodramatic way.

We also use the phrase 'hamming it up', which means 'to over-act' and comes from a trait common amongst younger actors to overdo the drama a bit when playing the role of Hamlet in Shakespeare's eponymous play.

In the case of the soccer player I might suggest the word 'baiting'. Perhaps this could describe the Police Officer too however, there is the element of controlling the situation when a Police Officer does this - they are establishing a power dynamic; whereas the soccer player is just hoping for a positive outcome.

The word 'juking' also comes to mind.

While the question is about exaggerating a response, the examples given seem to be more about active deception (particularly in the police officer example). A few words could help explain:

feign

verb (used with object)
1.to represent fictitiously; put on an appearance of:
to feign sickness.

2.to invent fictitiously or deceptively, as a story or an excuse.

For example,

The soccer player was feigning his injuries.

Dissumulate

verb (used without object), dis·sim·u·lat·ed, dis·sim·u·lat·ing.
2.to conceal one's true motives, thoughts, etc., by some pretense; speak or act hypocritically.

For example,

"Woah woah woah, calm down", the police officer responded, dissimulating.

See also dissemble.

Additionally, a charade, can often involve using exaggerated actions or appearances for the purpose of deception.

3. a blatant pretense or deception, especially something so full of pretense as to be a travesty.