This is a great question. The larger issue that you've identified is sometimes referred to as Periodization. Basically, where do we draw the lines between different periods? Can we draw them? What do we call them? This is certainly something that gets talked about a lot by all sorts of historians. For a great discussion of the issue as it relates to the Middle Ages and the early modern period, I would point you toward Constantin Fasolt's Hegel's Ghost: Europe, the Reformation, and the Middle Ages which is handily available as a PDF online, but which was originally published in the journal Viator.

The specific question you're asking is about terminology, and you've correctly identified some of the issues with the terms that we've traditionally used. The Middle Ages implies some dreary time of violence and no progress stuck between the Classical Period, or Antiquity, in which everything was better, and the Renaissance, which was the Rebirth! the Return to all that was Good! There are definitely some value judgments embedded in those terms.

You've probably heard that the term Renaissance actually comes from that time period. The humanists considered themselves to be causing the rebirth of European civilization, and gave it that name, at the same time slighting the Middle Ages. That's the over-simplified version taught by 7th-grade teachers, but it is roughly true.

In the past 30 years or so, historians of the Middle Ages and the Renaissance started questioning where the lines should be drawn between the periods, and whether they should be drawn at all. The upshot of the debate is that we've begun referring to the Renaissance, which didn't really work as a description for all of Europe during any one unified time, as the early modern period. The boundaries are roughly 1500 to 1750, with fuzzy borders at both ends. The Renaissance, if historians refer to it that way, is the movement of Italian humanism, not a time period itself.

The term the Middle Ages, or medieval period, has stayed. To be completely speculative, I think it's partly because we medievalists secretly like it, especially medieval (we can spell it with an a, and then it's Mediaeval and it looks really cool). It's also really hard to kill terms, both academically and with lay people (academic historians haven't really referred to the Dark Ages...in at least fifty years). So, we've kept the Middle Ages, and divided them into three parts. The early Middle Ages, which run roughly from the fall of Rome to the eleventh century, the central Middle Ages (sometimes called the High) from the twelfth century to the beginning of the fourteenth century, and the late Middle Ages, from the fourteenth century until the beginning of the early modern period.

Your instinct that these divisions are arbitrary is spot on; but despite that, “the Middle Ages” is probably here to stay, just like “classical antiquity” and “Renaissance.” They’re too ingrained in the shorthand historians use to be dislodged at this point. However, all three are being reevaluated and fine-tuned. As to how the “Middle Ages” got its name, we have to go back to the Renaissance, which, in a self-congratulatory way, named itself. Since it saw itself as reviving classical antiquity, it had to characterize what came between it and themselves.

People like Petrarch had already bemoaned the backwardness of their age. But more precise periodizations began to take shape with scholars like the papal secretary named Flavio Biondo, who wrote a work published in 1483 called The History of the Decline of the Roman Empire, which he dated to 410. How to characterize what came after that? It was all basically darkness and ignorance. The Italian artist and art historian, Giorgio Vasari (a student of Michelangelo), wrote a history of Renaissance art first published in 1550 in which he used the term “gothic” to describe medieval art. He meant that it looked barbarous, like the art of the barbarian Goths. He called his own age a “renascita,” a rebirth. Finally, in 1604, a German legal historian--Goldast--coined the phrase medium aevum. This phrase became widely accepted with the publication in 1675 of a work (a textbook, really) called The Nucleus of Middle History between Ancient and Modern written by a Dutch schoolteacher named Christopher Keller ("Cellarius"). (Sir Henry Spelman first used the term “the Middle Ages” in English in1616.)

The battle lines were hardened by the Swiss historian, Jacob Burckhardt, who published the very influential The Civilization of the Renaissance in Italy (1860). He described it as a great turning point, a rejection of medieval “barbarism” and “superstition.” His work (along with that of the French historian Jules Michelet) popularized the term “Renaissance.” Medievalists soon answered back. For instance, in 1927 Charles Homer Haskins wrote and important book called The Renaissance of the Twelfth Century where he argued that an intellectual revival occurred a full century or more before the traditional start of the Renaissance (c. 1300). Since Haskins, we have very much refined our understanding of the roots of the Renaissance, which was an evolution of trends already visible in the Middle Ages. In fact, your traditional use of Classical, Medieval, Renaissance would be adjusted by most historians. For instance, 40 years ago Peter Brown introduced the idea of a “late antiquity” (from the growth of Christianity to the 500s or so) to describe the period where the imperial Roman ethos fused with the Christian one. Your observation that “it would be more correct to say that the Renaissance begins at roughly Time X, but the Middle Ages continue in parallel until Time Y” is in fact how the two eras are understood now. Many historians prefer to speak of “Renaissances” since it characteristics mature at different times in different places, most obviously in Italy compared with its later arrival in northern Europe. Likewise, the beginning of the Renaissance, often dated to Petrarch (1304-74), is now widely acknowledged to have its roots decades before him. Thus, Ronald Witt emphasizes the importance of scholars like [Lovato de’ Lovati] (http://en.wikipedia.org/wiki/Lovato_Lovati) as early humanists. He died in 1309.

Finally, there has never been consensus about when the Middle Ages began and ended. Some traditional starting dates: 395-Death of Emperor Theodosius, 410-Sack of Rome, 476-Fall of Western Roman Empire, 590-Papacy of Gregory the Great, 800-Coronation of Charlemagne. Some traditional ending dates: 1453-Fall of Constantinople, 1450s-invention of printing, 1492-Discovery of New World, 1517-Lutheran Reformation. Several reference works say "1500"--"presumably on New Year's Eve" as Fred C. Robinson puts it. Some go even later” 1789- the end of feudal contracts in France!

Whatever the case, since the Renaissance, what we call the “Middle Ages” has always been used by other eras to help define themselves. As historian Brian Stock puts it: "The Renaissance invented the middle ages in order to define itself; the Enlightenment perpetuated them in order to admire itself; and the Romantics revived them in order to escape themselves."

Sources: Brian Stock Listening for the Text: On the Uses of the Past for the evolving idea of the Renaissance. For early humanism, Richard Southern, * Scholastic Humanism and the Unification of Europe* and, especially, Ronald G. Witt, The Two Latin Cultures and the Foundation of Renaissance Humanism in Medieval Italy.