To be sure, that map doesn't have a scale bar. If it's following the same scale as the
Star Charts map that's used as its basis, one of those squares would be 20 ly per side. That's the size of a "sector" as first defined in the TNG Tech Manual...
The Rigel in that map ain't the Beta Orionis Rigel, because that would be "south" on the map, not "down" or "up". But the sky is full of real Rigels - in fact, Alpha Centauri is one of them (Rigel Centauri or Rigil Kentaurus, literally "Leg of the Centaur").
As for "star named Orion", that's a Trek thing... In TOS and TAS, there's mention of an ancient culture of Orion, and the TAS episode "Yesteryear" makes it sound as if the place is called Orion, too, not just the culture. In DS9 "Little Green Men", there's definitively a place called Orion: Quark's going to make a side trip there.
The question is, what sort of a place is it? A constellation doesn't make sense as a waypoint for Quark, because one can't make a "side trip" to a dozen stars scattered all across the sky. A star or a planet are better alternatives.
Whether the star/planet Orion is related to the Earth constellation Orion in any way is debatable. Might be it's named after the Orion civilization instead, and the name of the Orion civilization in turn doesn't come from the Earth constellation but is instead just coincidentally the same word... Much like Klingons aren't named because they resemble cling-ons (at least as far as we know).
Of course, some fans of old saw the contradiction even back when TOS and TAS was all we had. Instead of yielding to the idea of an ancient civilization named after an Earth constellation, they decided our heroes were in fact speaking of "O'Ryan's planet"...

The good old
Star Maps refers to this one, in addition to featuring stars of the "real" constellation (which incidentally include one of the Rigels).
Timo Saloniemi