What is the precise difference between "wince" / "flinch" / "grimace"?

I would be sceptical of any site that defines anything as synonymous with "shirking away", since that phrase makes no sense in English. "Shirking" means avoiding a duty, and is unconnected with facial expressions.

"Grimacing" is all about facial expression, basically on the spectrum between disapproval/dislike/disgust/revulsion.

The thing shared by wincing and flinching is that they are basically pain responses. Almost the same but not quite. One nuance is that wincing is more about actual pain received (pain here subsumes the mental sort, people even wince at bad puns), whereas flinching is about avoidance of anticipated pain. In some contexts it is considered cowardly, so that "unflinching" becomes a word for courage and determination, sometimes ruthlessness, whereas if there is such a thing as "unwincing", it's less common and a lot less likely to be heard in e.g. patriotic or revolutionary rhodomontade.

You ever had someone touch your eyeball? You can let them do it without flinching? It's not easy.

There is no obligation to have any facial expression when flinching. The body part under threat is withdrawn, that's all. Put your hand on a hot-plate, and you'll surely flinch, you may wince as well if it burns you. You don't have to grimace.

(I see this overlaps on Josh, who wasn't there when I started to compose)

Since you've looked through various source material, I'll avoid throwing dictionary terms at you and state this as I see it, which may not be how others see it. But I will involve the supposed etymology of the words.

1. Flinch - If I act like I'm going to punch you and you flinch, that means you moved your body in a twitch-like manner, in this instance away from me. The same applies if someone jumps from behind a bush and says "boo!" It is probably from the Old French flenchir - "to bend". Etymology of Flinch

2. Wince - I usually think of this as a facial reaction as far as modern usage. In my thinking it involves squinting or averting the eyes while making a grimace, and is often an involuntary reaction to something unfortunate that has happened to someone or something in front of you. Before the Anglo-French winchir (to recoil suddenly) it was probably from the Old French guenchir - "turn aside" - and may involve the body, hiding the face or turning the head, but again I would think that's secondary to the facial expression. Etymology of Wince

3. Grimace - supposedly traces back to Old Saxon grima - "mask" - and is purely a facial distortion - it is a caricature of disgust, it can be similar to a wince or even used purposefully to "make a face" at someone. Etymology of Grimace

My opinion-

A wince is a facial or bodily expression of pain, disgust, or regret.

A grimace is a facial expression that usually suggests disgust or pain, but sometimes comic exaggeration.

To flinch is to pull away suddenly or recoil when something frightens or hurts you.

• he was so tough, I thought he'd never flinch.

Josh61 already called this out in a comment, but since you ask:

What are the exact differences between the three terms, and in what situations are they best applicable?

I wanted to call more attention to another, metaphorical meaning of flinch.

1.1 (flinch from) Avoid doing or becoming involved in (something) through fear or anxiety:
I rarely flinch from a fight when I’m sure of myself

_{"flinch" Def. 1.1. Oxford Dictionaries, Oxford Dictionary of American English, n.d. Web. 8 May 2015.}

In this sense, flinching is commonly used in the context of backing down, relenting, surrendering; to balk. It can be used when you are described a meta-confrontation, like a negotiation, a business or diplomatic deal of some kind. It can be used to describe behaviors anthropomorphized entities like countries, states or corporations.

Wincing is a poor synonym in this context, and grimace I have never heard used to describe anything other than a bodily reaction. You would have to construct a more complicated metaphor to describe an organizational wince or grimace.