Is this a cash bar

No that isn’t a cash bar. A cash bar is when guests pay for their drinks.

It’s not a cash bar unless you will be charging your guests for their drinks.
Which you shouldn’t because you should be providing drinks as the host.
If you don’t have license to sell alcohol it could also be a legal issue if you were to charge guests. All states are different and I don’t know all the laws.

Oh no we’re not going to charge our guests

It’s not a cash bar then if guests don’t have to buy anything

No. A cash bar is where guests are charged to drink.
A cash bar is considered rude because you don't invite guests to your home for a party or dinner you are providing and tell them they have to pay for each drink. A wedding is just a party on a larger scale but the same etiquette applies.

It's only a cash bar if you're not paying for the alcohol at all & guests will be purchasing their own. People consider it rude because they feel you should be providing your guests drinks if you're going to bother having a bar. I don't find it rude at all especially since alcohol can be pricey & some people would rather not pay for anyone's drinks. We're having a cash bar because neither of us drink & hosting a bar is expensive, it didn't fit into our budget & that was the thing we really didn't think was important.

No, this is common. Bars are named from the guest's perspective, so a cash bar is where the guests have to pay for their own drinks. That is frowned upon in some circles because your guests shouldn't have to pay to attend your wedding.