Well, to take one question, someone who has a trust fund which will steadily pay out enough money for middle class class lifestyle for the rest of their life is probably "wealthy" but not rich (unless of course they set aside enough and wisely invest from that regular payout, or otherwise go beyond the obvious scenario).
The definitions used by wealth advisors:
Richness is relative. If you make $25k a year in NYC, you are not rich. If you make $25k a year in Zimbabwe, you are quite rich.
Richness is sustainable. If you make $10k a day, you are rich regardless where you live. If you make $10k in one day and nothing the rest of the year, you aren't rich, even on that one day.
Richness is based on what you can afford, not what you choose to have. If you have $500 million of assets and you drive a Ford and do all your shopping at Walmart, you're still rich.
In summary, you are rich when you can sustain a lifestyle requiring significantly more money than an average resident of your area.
"Significantly" and "area" are flexible and somewhat subjective.