Word Usage – Difference Between ‘The Third Time’ and ‘For the Third Time’

It's an interesting question. As @Mistu4u mentioned, the formal definitions of both words are similar.
There are some differences, however:

Happen — something random to take place;
Occur — something to become observable; in a certain moment; an event or a process;

The difference is even more evident if you notice that occur is relative to current as they both came from Latin currere "to run".

So, I would use occur in the following contexts:

• Formal conversations;
• When a time or place of an event is specific;
• Expected events, like dates: The Spring Bank Holiday that would usually have occurred at the end of May;
• Discoveries, ideas: The discovery occurred in Mono Lake, California;

...and happen in the following contexts:

• Informal context;
• Rough time or place;
• Random events, usually without being planned;

As usual, you can step off these recommendations if you need to convey your attitude about what you are speaking about.

If you had taken up woodwork as a hobby and made your first object, lets say a wooden jewellery box, and someone asked you:

That looks beautiful. Did you make it?

You would proudly assert:

Yes, I did actually. I made that box with my (very) own hands.

Note, you could (idiomatically) insert very in the phrase without changing the meaning. The fact you place the adjective, own between the possessive adjective, my, and the noun, hands, means you are emphasizing the ownership of the hands that crafted the object. If you had replied:

I made that box with my hands.

Your listener would understand that you were the carpenter or maker of this object but it would sound odd, because you can't make an object with someone else's hands. You would never say: "Yes, I made it with his hands". It sounds not only gruesome (!) but obviously illogical. More typical expressions would be:

• Yes, it's handmade
• Yes, I made it by hand

The above expressions tell your listener that the object is crafted by hand, that very little machinery was involved in the process or if that were not the case, that you handled the machinery or mechanical tools yourself.

A more unusual way to express the same idea would be to say:

• I made it with these very hands

I would place the stress on "very" and probably hold my hands, palm upwards, to show the various cuts, nicks, scars and calluses that might be present as evidence that I am an experienced carpenter.

The Google Ngram seems to be heavily in favour of "by hand" but please do look at the different links below the chart to see how each expression can be used.

EDIT: Very when it is used as an adjective

There is a very famous and beautiful song sung by Nat King Cole entitled

The Very Thought Of You

The first two lines are

The very thought of you and I forget to do
The little ordinary things that everyone ought to do

In these lines, "very" means mere, simple, and plain. Just the simple thought of his loved one is enough to make him forget about the dull and insignificant moments of his daily routine. The meaning is slightly different from that quoted by the OP, "precisely as stated or being exactly the same and not any other"

However, returning to the OP's sentence,

Yeah! Trust me. I saw her with him with my very eyes

in this example very carries another different meaning. The Chambers Dictionary gives the definition for "very" used this way

adj. used for emphasis, [...] precise, actual (this very minute, her very words)

Thus the speaker is emphasizing that he saw the woman with a man, presumably not her real partner, with his actual eyes (in other words he didn't hear the news from anyone else) and therefore, to take his word for it.