As per the Memory Alpha article on the Miranda class, here:
http://memory-alpha.org/en/wiki/Miranda_className
The name Miranda-class was not mentioned in dialog on screen, but was devised by the art department of TNG to refer to this design. The name was used on several pieces of on screen signage referring to these vessels, and is also used by official references such as the Star Trek Encyclopedia.
and
Apocrypha
After its 1982 premier in Star Trek II: The Wrath of Khan, fan produced blueprints named this ship type the Avenger Class Heavy Frigate. Further non-canon fan publications created its mid-23rd century predecessor the Surya-class. Ships of the Starfleet suggests that the prototype vessel USS Miranda had the registry NCC-1833.
Some sources have described the class as the Reliant-class.
Fanon speculation by people wanting to keep the "original" Avenger designation say that the Avenger class is a heavy frigate, with her internal layout more geared towards tactical applications, like the hanger bays being host to 1-man fighter squadrons, sensors being more limited in terms of science packages, and other facilities of a more military bent. Thus she is more suited to border guard and neutral zone duties.
The Miranda class is a general purpose cruiser in the same hullform as the Avenger, with internal space configured with VIP quarters, cargo bays, as many labs as a Constitution, more versatile science sensors, standard shuttles with maybe a few special variants (cargo, aquatic, etc). As such she is the workhorse of the fleet, running colony resupply missions, ambassadorial transport in less strategically important regions, long range or long duration science missions, etc.
As to the mega-phasers, since we saw the
Reliant in a battle the Avenger was classified as a heavy frigate (despite the fact that she was on a science mission). Fanboys and gun-bunnies seem to forget the Federation is a peacenik operation despite their armed ships and are constantly assigning hugely inappropriate amounts of firepower to Fed ships.
A quick examination of the battle itself shows the
Reliant's "phaser cannons" inflicting pretty much the same level of damage as
Enterprise's standard saucer phasers, so there was no real reason to call them "mega phasers", IMHO.
My take on that is, since the Klingon heavy disruptors (the green crap a D7 fired from its nacelles in
'TOS: Elaan of Troyius') can be fired while the D7 was at warp speed, and phasers are supposed to be lightspeed-only, the "mega-phasers" was Starfleet's way of pushing standard phasers into warp-speed phasers. Pure conjection on my part, but take it for what it's worth.