Well, it just depends on what you're goal is. In your situation, it sounds like you should shoot ANY buck that comes along, or any doe (if legal to do so). Or you might want to let the younger/smaller bucks pass on the hopes that another hunter won't shoot it and it will be bigger next year (or the next).
To my way of thinking and understanding, a "management deer" is one that is killed to enhance the health of the herd overall, OR to cull out undesirable genes.
So a "management doe" is just ANY doe if the ratio is out of whack, and does need to be killed to help restore a better ratio of bucks to doe, or just to reduce deer population in general, if it's overpopulated (to prevent disease, ticks, DVCs, etc.). I always shoot does if I can and have a tag for one, because our ratio is out of whack, and the ratio being out of whack makes it much harder to get trophy bucks, because they don't have to compete for does, so things like mock scrapes, grunting and rattling don't work as well on bigger bucks.
A "management buck" is one of ANY age which has deformed antlers, if you are trying to promote large nice typical antler genes. So to me, no, just being an old deer doesn't make it a management deer, and if fact, how are you gonna tell anyway hold old it is just by looking? You can't. Besides, an OLD buck is gonna have large antlers, so you're going to shoot it anyway, because that's what you're looking for. So an old deer is not killed *because* it's a "management buck"; it's good because it's a "good buck" - but either way, you take the shot if presented.
If it's old AND just happens to have deformed antlers, then yes you'd want to shoot it, but that comes under the heading of "shoot any deer of any age with deformed antlers".
But on private land, or on public land in spots where very few hunters go, if the antlers are not deformed, then the rule is simple: let the bucks walk, unless they're big enough to be a trophy TO YOU. That will change over time as you harvest bigger and bigger bucks.
For me, right now, with a BOW, I would and will shoot any buck that I think is 2.5 years old or older, with an 8 point or better rack that is more than about 15" across. But with a gun or muzzleloader, I'm not going to shoot a buck unless it's a true trophy (to me) - at least 20" across and big, thick, long antlers.