Is there a real name for "local celery"?

I never knew there were different kinds of celery until I came to Trinidad and locally grown celery looks nothing like USA grocery store celery(recently become available imported), it is small and deep green and sold leaves and stalks. It is very tough but has a VERY intense flavor and smell, both the leaves and stalks are used as a flavoring instead of a veggie, I don’t even think you could eat it like USA style celery.

Now clearly some research tells me this is available in the USA:

However it is also unhelpfully named local celery too.

Is this just young celery or a different kind or what?

At first I thought you might be referring to Chinese celery, but I think what you have is not a different variety of celery. Celery must be given plenty of water and be grown in cool weather for it to develop large, tender stalks. Another thing that contributes to tenderness is blanching - protecting the growing stalks from light by either mounding dirt around them or putting a cardboard tube around the plant. It looks to me like your “local celery” was grown in warm weather, with full exposure to light, and without a lot of water.

No I think you might be right, this picture here looks exactly like local celery: