What is vested monthly benefit in a 401k mean?

That means that the money your employer is matching and putting into your 401K is not available all at once. More and more of it becomes yours the longer you work there. Although the money shows in your account, if you leave the company, the "vested" amount is all you are entitled to keep. If you roll the money over into a new employer account, your former employer will keep the portion in your account which is not "vested".

It usually goes by years you've been there rather than months. You could have to work there 5-10 years to be entitled to all of the matching funds.

Vested means you have you made it past the waiting period, usually 5 years, and you can use your money if you want. I wouldn’t til you retire,but you could.

When you open a 401k, most employers will give you an additional sum to put into your account every pay period.

In most cases, if you leave your job within a certain period of time (usually 5 years), the employer removes all or some of his contribution from the monies you have in your account.

After that period, the employer's contribution is kept in your name even if you leave your job. This is vesting.

The amount you put in is always yours. So your contribution is always vested.

it means the amount quoted is the definite amount of distribution when you choose to take it, you can request more but the vested is a given, cannot be less

You become vested in your company's contributions after so many years. Not always, but generally:
1 yr 20%
2 yr 40%
3 yr 60%
4 yr 80%
5 yr 100%

That makes no sense to me. I've not heard of a 401K that guarantees a payout benefit. Did it say balance?