So, this concept of prehistory being defined by the introduction of writing is sort of outdated in the field of historiography. It used to be thought that nothing could really be known about the past without written records, so therefore history begins with the introduction of writing, and everything before writing is "pre-history" and outside the realm of the formal academic study of the past known as "history."

But over the last century or so, historians have gained a more nuanced understanding. We've come to learn that some oral histories are actually quite accurate and have been corroborated, and the advanced excavation techniques of archaeologists and anthropologists have enabled us to learn a lot about people in the past who left no written records and many times left little material culture at all. So historians no longer necessarily say "history began around 4000 BCE when the Sumerians invented writing."

All of that being said. What is known in any significant detail of history does mostly begin with the introduction of writing in various places. Writing is generally independently developed in societies that have grown to become complex enough that face-to-face relations between people who are acquainted with one another are no longer the sole way people interact. Which is to say that writing develops when a culture begins to urbanize and develop a division of labor and therefore an economy. In most places, such as in ancient Mesopotamia, the oldest known written records are tax and debt records.

Obviously, this means certain parts of the world got over this period at different times, some by inventing their own writing systems, and others by getting it imposed by outsiders (Like the Americas, for example). Also, that some civilizations made a "jump" or a "flash forward" of thousands of years by going directly from prehistory to the 1700's, in the case of Australian natives.

You are making the mistake of assuming that History is progressive and linear. It is not. No people went "from Prehistory to the 1700s" because "Prehistory" is the absence of writing, and "the 1700s" is simply the time between the year 1700 and 1799 AD. What happened in this case, was that people went from Prehistory to History (or better said, "entered History"), and that it happened in the 1700s (and that would only be true for Aboriginal societies who adopted literacy, or became integrated into European society, which I suspect actually happened later). "The 1700s" is not a technological or cultural level. It is just a portion of time.

So, I wanted to ask what were the main instances in which a civilization came out of prehistory by adopting a writing system, regardless of whether it was foreign or of its own invention, or if it was imposed or adopted voluntarily? I know the discovery and gradual conquest of America and Australia are among the main ones, but what about other parts of the world like Europe or Asia, when did prehistory end in this places?

This is such a large topic as to be impossible to answer in a single post. In Europe, for example, the first script used was Linear A in Crete and parts of continental Greece from roughly 2500 BCE to 1500 BCE. It was succeeded by a different script, Linear B, which was used until the advent of the so-called "Greek Dark Ages" between roughly 1100 and 800 BCE. During the Greek Dark Ages, writing was lost in Greece (and as a result, Greece entered a centuries-long phase of "Prehistory" after centuries of "History"), and the later Greek alphabet which was adopted after 800 BCE and has reached our days with some variations is completely unrelated to both Linear A and Linear B. Instead, the Greek alphabet was an adaptation of the Phoenician alphabet developed in the Levant around 1200 BCE. So from roughly 1100 to 800 BCE, the whole of Europe would be in "Prehistory". After that, "History" would be present in the Phoenician colonies in the Mediterranean area, the Greeks and their own colonies, and a number of Mediterranean peoples that adopted writing from either the Phoenicians or the Greeks, such as the Italics and the Iberians. For a good chunk of western, central and northern Europe, the arrival of writing and therefore, the beginning of History, is a consequence of the Roman expansion, either directly or indirectly (in the case of the Germanic runes ). For eastern Europe outside the Balkans and Crimea, writing was introduced after the 9th century AD by Byzantine missionaries in the form of the glagolitic and cyrillic scripts.

Also, I'd like to know if there's another element that is used to mark the end of prehistory in an area other than the arrival of writing.

Such thing is impossible because History is literally, the study of writing records.

However, you might find interesting the term Protohistory, which is applied to cultures that have a state structure but no writing, or that are known through the writings of other cultures they were in contact with, but had no writing of their own.