To directly answer your question: There is no "precise" definition of the term "supermoon."
Term Origin: The origin of the term is generally attributed to the astrologer Richard Nolle in 1976. As Wikipedia notes, "In practice, there is no official or even consistent definition of how near perigee the full Moon must occur to receive the supermoon label, and new moons rarely receive a supermoon label."
Term in Practice: In practice, the term is usually used when perigee (moon's closest approach to Earth) and syzygy (alignment between three bodies -- in this case, Sun-Earth-Moon) occur within about a day of each other. Also often in media practice, the term will usually be applied when perigee and syzygy occur closest for the year, and/or when perigee is closest for the year and it's within a day or so of syzygy. An added qualifier is that the term is practically never applied to the new moon, only full moon, despite the new moon also being a syzygy point.
Meanwhile, at TimeAndDate.com, they use the definition of a supermoon being when "a full or new moon" happens when the moon is also less than 360,000 km from Earth, so they at least have tried to give a numerical definition though they don't define exactly what a new or full moon means, since a true new or full moon only happens during an eclipse.
How Much Bigger? The apparent size of the object in our sky is directly proportional to distance when angles are small (numbers from Wikipedia). The average perigee distance is 362,600 km, and average apogee is 405,400 km. Simple division indicates that the average perigee moon will be ≈12% larger than the average apogee moon. With a semi-major axis of 384,399 km, the perigee moon would only be ≈6% larger than an average full moon.
How Much Brighter? Here's where my answer varies from the other, which is incorrect. Brightness follows the inverse-square law, meaning that brightness falls off as the inverse-square of the distance. So, we have to square the result in the previous paragraph: (1.118...)2 ≈ +25% brighter. So, the perigee full moon will be roughly 25% brighter (not 5%) than an apogee full moon. It will be ≈12% brighter than an average full moon. At the extremes (perigee 356,400 km, apogee 406,700 km), the perigee moon will be ≈14% bigger in the sky and ≈30% brighter than the apogee. So, yes, this is quite noticeable, and is a problem for astronomers who rely on dark skies. Telescope time is usually apportioned factoring in the lunar phase, and it is especially an issue when perigee and full moons align.
Other Names: To answer your last question, so far as I know, there is no term for a "closest perigee," and there are no special tide names for it. We do have spring and neap tides, but those refer to when the sun-earth-moon are aligned vs at right angles.