English term to describe "kompot", an Eastern European beverage made from berries

UK English would call it a cordial. Though typically strained as below:

Place 300g mixed fresh or frozen berries, 1½ cups white sugar and ¼ cup lemon juice in a heatproof bowl. Add 1 cup boiling water. Stir until sugar has dissolved. Strain cordial through a sieve, pressing down with the back of a spoon, into a large heatproof jug. Mixed Berry Cordial Recipe | Woolworths https://www.woolworths.com.au/shop/recipedetail/4123/mixed-berry-cordial

I would call it a raspberry pressé as in this recipe or raspberryade as in this recipe. Pressés and drinks with names ending in 'ade' are most usually sparkling these days but both terms originally to still drinks.

However your recipe seems a bit weak to me, all the recipes I've seen have a much higher ratio of fruit to water as do the ones under the links.

The word for your drink is a kompot. Not many people in English speaking countries have heard of it, since the preparation is rarely made, but it is the correct one. It is typically served cold, but there are people who like it at all temperatures.

It is not to be confused with compote - the name is probably etymologically related, but in practice, the compote has much more sugar and fruit, while a kompot is predominantly water.

There is a point to be made that this is not an English word yet, since most people haven't heard of it. Lacking another word that is more well-known, if you take this point of view, the conclusion is that there is no such word in English. I prefer to see it as a real loanword in English, which simply happens to be known to a very limited circle of people yet - words like "regmaglypt" are known to very few people and are still recognized as actual English words.

Here a Venn diagram of what your possibilites are. In my experience, when you cannot meet all criteria, using the rare word (and explaining it) is the best choice. Using the overly broad one (berry drink) is worse, because people imagine something entirely different and are misled. Using the incorrect word is the worst choice.

The unspecific and incorrect choices are worth considering when it doesn't matter that people are being misled, e.g. you are translating a movie dialogue and a character asks a friend to pour her that drink, and it doesn't matter for the plot what she drinks. It is still irritating if it happens to take a life on its own though, like "let them eat cake" (the original is "brioche", but the word was unknown in English earlier and the translator chose the incorrect "cake").