I also lived and hunted in Europe for 5 years and I never heard heard a single hunter refer to a Moose as "Elk" I hunted with many Germans who hunted in Scandinavia and many Swedes/Norwegians they all called Moose "Moose"
North American Elk are called either Elk or Wapiti, both names are correct
A bit of deductive reasoning is also useful.
1) I think we would both agree with the fact that Indo-European peoples have inhabited Northern Europe for many thousands of years.
2) I think we would also both agree that many of those people are of the Germanic linguistic group (apologies to the Sami, Finns, and Rus) and these people occupied part of Scandinavia which is a native environment of the animal we in North America call a "moose".
3) We know that these people hunted this animal for many thousands of years, and that the modern antecedents of their current languages began to emerge at least by the time of Julius Caesar.
4) I think we can both agree these Germanic peoples had a name for the large palmated antlered deer they hunted.
5) It is well established that the linguistic derivation of the word "moose" comes from the Algonquian group (the smarter members of the class can see where this is headed and are excused to recess). The Algonquian peoples are a native American culture who were found across the modern day northern US and much of Eastern Canada.
6) Earliest meaningful contact with the Algonquin peoples occurred 350-400 years ago. The first reference to the term "moose" in English is found correspondence of the early 17th century.
7) Either northern Europeans called the large deer they hunted "the animal with no name" for many tens of centuries awaiting the discovery of the new world, or they called it something in their own languages.
In German the word is "Elch". In Norwegian it is "Elg". In Swedish it is "Alg" with an umlaut on the "a". (Do we see a Germanic pattern?) The Finns, a non-Germanic people, call it a "Hirvi". I see you have hunted in Poland - they call it a "Los".
No one is entirely sure how English speakers first started using the term "elk" for wapiti. A general consensus that makes sense to me is that early English colonists quickly encountered Eastern woodland wapiti, knew it wasn't exactly a red stag, and so named it after the only other large deer of which they had ever heard - the "Elk" which is the anglicized form of Elch or Elg. When they eventually did encounter the real Elk, that name was taken in the vernacular, and so the native American name "moose" was adopted.
And yes, it is perfectly alright to call cervus canadinsis either an elk or a wapiti in North America. Many prefer wapiti so as not to confuse it with the European elk.