The rules on teleportation do vary, as different methods do have different conditions and limitations. The highest but most rare form of this would be 'true' teleportation, in that the caster can teleport anywhere with perfect accuracy and intent, with little to no limits on distance. But that method is so rare as to be considered a myth by most, and would be something that a prodigy of magic could spend their entire life devoted to the research of without achieving it.
What most teleportation experts would recommend, instead of that, is to have what they call an anchor - a destination in mind that's reasonably close, well-traveled, and comfortably familiarized by the individual. When those conditions are met, and assuming the destination is within a few hundred meters, what's known as a 'warp' can be attempted, should the person also know and perfect the casting motions of the warp spell. Warp itself is a decently difficult spell to acquire information on and practice, as many tomes on the spell would be kept under lock in magical libraries. Even were a caster to learn it, it typically requires multiple minutes of idle concentration to land perfectly; to arrive at their intended destination instead of inside a wall or floor. The average person can typically expect to successfully warp once, maybe twice a day before they risk rebound or an imperfect transportation.
A weaker variation on this would be the blink spell, which can allow one to warp short distances of about five meters at a time to a place within eyesight. The limitations put upon it make it easier for casters to control, though another drawback of its quick cast time is that users report a feeling of winded fatigue as if they had full sprinted double the length of where they had blinked to. As a result, it quickly becomes exhausting with repeated use in a short time, and is considered inefficient for that reason. It also uses up more essence than something of that scale would reasonably demand, to compensate for speed. Typically reserved for emergency scenarios to flee from imminent danger.
The last and strongest known primary method of teleportation would simply be the spell named 'teleport'. Its tomes reside within forbidden libraries across the world, guarded heavily. Knowledge of the spell is only trusted with respected Arcmages, or an equivalent rank in the hierarchy of other societies, and it's illegal for those who know it to teach it to those who don't meet the qualifications. There's not much public knowledge about the spell, but it's been said that at its peak, it allows for worldwide or even dimensional travel without limit.
More niche methods would exist, such as the shadow melding of practiced Umbralists, or the 'gate' spell taught to Arcmages and some Undermages, which opens a doorway to the location of a person they've essence linked with within two kilometers.
Teleportation rituals do also exist, but require setup and significant material sacrifice depending on the desired location. Alternatively, two-way teleportation altars can be constructed, though aren't considered practical due to cost and offerings necessary to maintain their power.
I might be missing something, but that roughly covers it, I think. I should also note that Rio didn't disappear until after the coin he had flicked had entered the water of the fountain. At the very least, after all the mechanical yapping I did above, it'd be safe to assume he didn't cast a spell, as no incantation or casting motions were made. Teleportation also simply dematerializes a person, whereas Rio's body almost seemed to become water and get sucked into the fountain. If it were a static A to B, it would likely fall under the category of a teleportation altar. Though in that case, while leyline alignment between both places would be encouraged for convenience, it isn't guaranteed.