A lot of the material and references you mention are part of the Dungeons and Dragons game source material.
In particular, Tales of the Sword Coast takes place in the Forgotten Realms setting, which is a campaign world designed by Ed Greenwood for his Dungeons and Dragons game, and which eventually became a licensed product that included source books, novels, video games, and more.
Dragon Lance is another licensed Dungeons and Dragons campaign setting, that was somewhat modified from the main rule sets available at the time.
Magic the Gathering, however, takes its cues from its own source material, although it does borrow rather heavily from other genres and sources, including Dungeons and Dragons.
In the basic Dungeons and Dragons original rules, Dragons were at the top of the chain of the various "draconic monsters". Each species of "true" dragon was identified by a basic color (red, blue, green, white, bronze, silver, gold), and had varying levels of intelligence, personality, and abilities (including the "breath weapon", which depended upon the color, and could be anything from fire to magical gas). Later on, the types of dragons were expanded consisiderably, including things like "gem dragons", planar dragons, and undead dragons (dragon liches). The size of dragons generally scales with age, with adults being quite large (dozens of feet from tail to snout on even moderate sized adults).
Some dragons are quite intelligent, surpassing that of most humans, while others are on the low end of the intelligence scale.
Dragons traditionally had a notable fondness for treasure.
Smaug, the dragon from Tolkien's The Hobbit, was undoubtedly one of the primary inspirations for the early Dungeons and Dragons dragon types. However, later on there were other dragon types introduced more closely based upon eastern mythologies, or purely from the authors' imaginations.
Drakes are sometimes used to refer to immature dragons, but are more frequently associated with much smaller reptilian animals that are mostly just 'scaled down' versions of regular dragons. Depending on the source, they may or may not be intelligent, and may or may not have the ability to breath fire. Typically, they are large reptiles (from 2-3 feet in length up to much larger sizes, depending on the source) that can fly and have a generally "dragonlike" appearance.
Wyrm sometimes refers to the oldest and largest types of traditional dragons. However, it can also refer to dragons that are specifically wingless.
Wyverns are quite separate. They are normally depicted as smaller than a full-sized dragon, but still quite large. They are winged and frequently have a barbed tail (sometimes poisonous).
However, as has been mentioned elsewhere, the terminology varies widely from source to source, and many of terms can easily be interpreted as interchangeable in many works.
** In traditional European heraldry, wyverns were typically depicted as dragon-like critters with only one pair of legs instead of two (or sometimes without legs at all).