Is this leaf celery/cutting celery?

I grew it once & that is exactly what I got. I used it for chicken stock.

LOL, this looks like what grew in my garden this spring.

There are three main types of celery: stalk (what you see in US supermarkets), bulb (more common in Europe), and leaf (more common in Asia). There are dozens of cultivars but here in the USA, the dominant one is 'Pascal' which is a stringless variety.

Yes, that implies that there are stringy varieties. I'm no expert on celery identification. No matter what it's called, it's in your hand and you're trying to use it.

I would peel the stalks and cut diagonally on a bias. They could be part of a stir fry or a soup/stew if you don't enjoy it raw. If you do eat it raw, you have the option of combining it with other veggies like shaved fennel or shredded cabbage.

Depending on the flavor/texture, sometimes the leaves and upper stems can be easily enjoyed, other times they might be more appropriate when making stock.

Asians pickle a very wide variety of vegetables and this might be a good candidate. Americans pickle a very small selection of vegetables. Depending on the texture, I might try doing a quick pickle like Japanese "asazuke" with this celery thinly sliced.

Best of luck.

Looks like Chinese celery to me. The flavor mellows a bit when it is cooked. There is a wonderful Sichuan dish called Send-the-Rice-Down Chopped Celery with Ground Beef, from Fuchsia Dunlop (in her cookbook, Every Grain of Rice), https://www.blossomtostem.net/sichuan-send-the-rice-down-chopped-celery-with-beef/

You'll need Chinese chili bean paste, called doubanjiang or toban djan. I don't de-string the celery since it will be chopped somewhat finely. It's my favorite way to eat lots of celery and it's sooo delicious & spicy.

I have never heard of cutting celery, but this looks like what I have occasionally seen in grocery stores sold as organic celery. Always smaller and the ribs do seem very tough.