Translation:We are not swimming in the river, but in the sea.
Ússzuk sounds pretty cute. :D
Whether you use indefinite or definite conjugation depends only on the direct object. That's the one the appears in accusative, in Hungarian with a -t suffix. Since úszik, like most -ik verbs, is generally intransitive, meaning it doesn't get a direct object, you won't see it being in its definite form.
(There are some constructions which lets you use definite conjugation, like "We swim the length of the pool", but those are rather rare. Ússzuk a medence hosszát, I would guess.)
Though often used interchangeably, strictly speaking our planet has seven oceans, and seas are parts of the oceans surrounded by land on most sides. There are also salt lakes with no connection to any of the oceans that are called seas as well. In Hungarian we don't use "óceán" and "tenger" interchangeably, but according to these definitions.
Some add the Arctic Ocean to get four, or the Southern Ocean to that to get five. That's also the number that the English and Hungarian Wikipedia entries use.
To get the traditional number of seven (as in "sail the Seven Seas"), Atlantic and Pacific are divided into North and South parts each.
Finally, some say there is only one ocean, the World Ocean, since all the ocean waters are connected and there is no obvious divide separating one from another.
Like continents (or, for that matter, planets), it's more a matter of convention as to how many you count, because it depends on how you divide up things.
With no context, the generic "tenger" can mean "sea" or "ocean." As an example, "tengerpart" would translate as "seashore," but this is used to mean a coastal (ocean or sea) shore. Look up these two Hungarian words on Google images and see all the ocean images and crashing waves that appear.