Is Irish buttermilk different from Estonian buttermilk?

Just to confirm what Joe said with some sources, the Estonian Dairy Association confirms that Estonian buttermilk or pett is a fermented product. That is, one takes milk and adds a culture of lactic acid bacteria (similar to how yogurt is made, though typically buttermilk is fermented with slightly different bacteria at lower temperatures, resulting in a thinner product).

Meanwhile, Irish dairies seem to refer to Irish buttermilk as derived from churning butter. And official documents and reports such as this one from Ireland define buttermilk as the following:

Drinking milk comprises raw milk, whole milk, skimmed milk and semi-skimmed milk consumed in liquid form. Buttermilk is a derived product remaining after milk or cream has been made into butter.

Buttermilk derived from butter making can vary significantly in its composition, flavor, texture, etc. Some butter is made by churning fresh cream, which produces a buttermilk with little acidity, while I assume much of Irish butter is made with cultured cream, producing the typical sour product that's superficially similar in some ways to fermented "buttermilk." But how much butter solids and fat remain in buttermilk made from churned cream is up to the manufacturer and the process. The color can also vary depending on that fat content, as well as the exact type of milk (breed of cows, feed, etc.).

It sounds like what you're getting is "cultured buttermilk", which as you noted, is not the leftover liquid from having made butter, but is basically runny yogurt.

I'm not familiar with Irish buttermilk, but it sounds like it's the older sense of the term "buttermilk". (ie, the milk leftover from making butter)