Quite an interesting discussion. As the original question is posed, it is a matter of nomenclature, so let's step back and consider how these terms might apply.
Now, "experienced" would mean, in normal practice, as being capable of reacting to a normal range of events within the area of interest, in a proficient manner. For those on a reasonably dedicated pursuit, a steadfast two or three year learning curve towards storm chasing, involving earnest field experience and a well-grounded understanding of the nature, evolution and behavior of severe thunderstorms would likely qualify a participant as experienced. As another poster pointed out, perhaps the more problematic portion of the definition is - what is a storm chaser? Does the ability to forecast the location and time of tomorrow's severe storm qualify? Well, single-minded pursuit and boundless energy can easily make up for a 200 mile forecast error 24 hours out. Just as with professional sports, a game-day elevation can overcome superior preparation - sometimes, but not always. So, any definition of storm chaser should be oriented to results, and stear clear of a bias towards forecast ability only. We live in a cyber age, but I would hope the community would generally accept that storm chasing involves physically chasing storms - bringing your own body to within proximity to a storm such that you are in a position to report unique and valuable observations based upon your own five senses. Now, the definition of chaser comes to a finer point, as involves the subject of photography. If we accept that a first-hand storm report of high quality represents the pinnacle of storm chasing, then how shall a quality photograph weigh against a just-as-true, but nontheless verbal report? Well, as the visual (and, at times audio) effects of this phenomenon of storms is quite descriptive, I would suggest that even an average, or most assuredly a well-done, photograph or video sequence, would carry quite artistic weight to the value of any storm report. The excellence of storm chasing should not boil down to photographic skills in isolation, but certainly an excellent photo contributes to an excellent report and should be given due credit.
Now, with regard to experience, I think also we must consider the importance of intellecual honesty. As I paused to consider this from my own experience, the thought dawned upon me: I've never actually been behind the wheel on a chase. I've been riding shotgun, reading maps, forecasting along the way, shooting pictures, etc., but over the next one or two years of learning, to consider myself an "experienced" storm chaser, I think I probably should actually be the driver one of these times! It seems that would be a visceral part of the chase. So, to answer one of your questions, Glen Romine - YES, to consider one an experienced storm chaser, I think yes, one must actually chase the storms! Man's conception of storms, man's tools used to measure and predict storms are one thing - but the storm itself is of overriding import, and to physically chase the storm is the essence of the hunter. So, while an earlier post suggesting "a season on your own" as one criteria might not be literally sensible, I think the spirit is well-taken: to assert yourself as a chaser, it is fair to expect you would have trialed all the mechanical and intellectual exercises involved in chasing. Perhaps not in isolation for an entire season, but at the very least entrenched yourself in each and every role. "I ignored the maps all day, but shot great photos when my partner got me there" or "I don't actually ever go out, but am a reliable nowcaster" are an entree; but not the description of complete chaser. Likewise, if personal or financial circumstances have prevented one from fully pursuing the hobby in initial concerted effort for 2-3 years, or if the atmosphere has proven unusually dry during one's chosen immersion, then honesty demands extending the annual dues.
The next question concerned the definition of "veteran." Unlike merely experience, for most careers veteran implies a participant who has been consistently at it for a good while - most likely at least half of his "career span" and more commonly, at least two-thirds. Normally, then, this wouldn't be much of a problem to consider, except for the unique fact that storm-chasing itself is relatively young. For this reason, it's hard to define what a normal "career span" might be. We read tales of the pioneers in the seventies, some towards retirement from the pursuit, but considering the influx, and the part-time nature of the pursuit for many of us, it's kind of hard to pin down the chronological point of a veteran to this emerging hobby, or even more elusive is how long the mass of us expect our collective career lengths to be. However, it seems the hobby has matured past the point where only the pioneers may be considered legitimate veterans. No hard and fast rule of thumb, but it seems that if one introduces themselves to me as having chased storms for 12 years or more, I would accord them the status of veteran.
Now, someone brought up the distinction between "storm chaser" and "tornado chaser" which must be given consideration. The broader definition cannot be given slight. However, when we think of storms, I believe it should be in the context of 'that which is worth pursuing.' I mean, the ratio of common thunderstorms to tornadic events must be quite high. A garden variety thunderstorm will hardly catch the attention of a domestic animal, so why should it be glorified in the human sense? That which is rare is to be glorified, and the tornado is the most prolific exemplar. To be included, however, would be spectacular lightening, raging flood, and mighty hurricane. With regard to flood and hurricane, the practical chasing problem becomes one of specialization, due to lack of opportunity. It seems, though, that a common sense standard can be applied, neither exluding non-tornadic storm reports from the fold, but maintaining tornadic reports as the exemplar of storm pinnnacle.
So, "experienced" should not be so much trouble for us to understand and accept in discourse. Those on the upside cusp will not be quick to claim the distinction before they are ready, those on the downside cusp will have an assured knowing which needs no artificial prompt. "Veteran" is a little touchy subject at the moment because the hobby itself is relatively young, but in time the distinction will naturalize itself. What would be approprite at the current hour, though would be the establishment of a "Storm Chaser Hall of Fame." This distinction would generate some real excitement, some real debate and also begin to establish a standard of excellence and achievement which seems to be a yearning, and rightfully so at this point. This is just a suggestion, any details would of course be the subject of a different thread. I think this pursuit has reached the point where individuals deserve recognition in the right of a chaser, and enough events have taken course such that honor and distinction is due.