Fallacy Name?

I come across a lot of fallacies when i talk to other people but this one has gotta be very common, and obviously fallacious, it's that:

Person 1 and Person 2 have an argument based on issue z, Person 1 is arguing in favour of defending z where as person 2 is arguing in the contrary.

Person 1: *makes argument x*

Person 2: *makes argument y*

Person 2: Argument y is a stronger argument for case z, therefore i win the debate.

The form of the fallacy here is that because something is better, therefore you are at a better position like "winning" when the central point hasn't necessarily been refuted.

This can be begging the question because there is an assumption of this when clearly not all instances being true in this form show it is a fallacy.

This can be a hasty generalization since it has not enough information to declare "winning the debate"

This can be argument from ignorance since person 2 assumes that his argument makes him win the debate so long that person 1 doesn't bring a stronger one, or refute the central point.

This can be false attribution since it just doesn't support what they claim.

but a problem with this is that at least 95% of the fallacies in the book can be considered "begging the question" since it's all based on assuming that your form of logic is valid or sound.

What do you guys think?

Well...that depends on what you mean by 'winning' a debate.

Some people would define it as being the better 'performer' rhetoric-wise. This is especially the case in TV debates, where sounding off clichés goes a long way in making you more memorable to the audience.

Others would define it as the one who makes better arguments. 'Better' here, refers to arguments that are valid and sound. 

Assuming someone made a stronger argument for their case, one could reasonably say the 'won' the debate (assuming there was a debate). But an argument being weak does not mean its conclusion is wrong - see the bad reasons fallacy. So even if person 1 made a comparatively weak argument, they may still be right - just for the wrong reasons.

(personally, I consider a person good at 'debate' if they can speak well, and good at 'argument' if they actually have a firm grasp on reasoning - the two are not the same!)

We could write what you're saying as follows:

1. Person 1 has a stronger argument than person 2 but the central point was not addressed.
2. Therefore, person 1 did actually addressed the central point.

The way I framed it sounds very contradictory and I think it's because there's a main driver that drives many fallacies depending on the issue. The main driver is wishful thinking. We could give two examples to see this.

Example 1:  Suppose there are two workers and have to finish a project by tomorrow. They also want to go to a concert that night. One of them, however, argues that they should stay while the other argues that they should go.

Person 1: We should go because we can relax and still finish anyways.
Person 2: But if we stay, we can order pizza and I love pizza!

We could say that person 1 has a "stronger" argument since staying to work because of someone else's preferences is not that great of a reason to do that. Going to a concert to relax sounds like a better strategy. But hard-working people might object by saying that both plans sound very risky. Staying for pizza sounds like not being that motivated and going to the concert sounds irresponsible. Here, person 1 "wins" but the central point has not being answered; priorities and time management. This is avoiding the issue if this is a response to the argument "your priority should be work." 

Example 2. 

Person 1: The reason why people haven't come to the bake sale is because there hasn't been any money coming in.
Person 2: The reason why people haven't come to the bake sale is because they do want to come, but someone is preventing them from coming.

Person 2 comes up with a conspiracy theory, and so we may think that person 1 makes a better argument even if the proposed theory has limited scope. The central point has not been answered: what is the reason why people haven't come to the bake sale?

We have two examples with two distinct fallacies, but those fallacies are committed because there is this wishful thinking of having solved the problem (perhaps out of a desire to win an argument). We could write it like this:

1. I wish I answered the central point.
2. Therefore, I did answered the central point.