I regularly encounter [1] and [3], but interesting to hear that [2] is “real”. Is it new? A random book from the 1990s I grabbed has sentences like, “When the critical value of the [Raleigh number], Ra = Ra[sub]cr[/sub] ≃ 657 is reached… the state with the fluid at rest in a horizontal layer becomes unstable…” A different book, from 2009, uses ≐ to mean “numerically close” and ≈ to mean “roughly equal”. (One might argue neither of those expresses “equal up to a multiplicative factor roughly equal to 1”, but if we are going to base it on asymptotic notation, why O instead of Θ or ∼ ?)
If I write “∼1000 ducks” or “π ≈ 3”, will I get told to go back to the 1950s?
Tilde is very common, but the two notations are not interchangeable. Tilde means something like “This is the actual number, and the limited significant digits I’ve given imply the level of precision you should assume.”
The other notation is a genuine “of the order of magnitude of…” estimation. Much less precision is intended, and it is used in contexts where additional precision would be erroneous or is actively undesirable.
“The stadium holds ~60,000 people.”
“Consider a stadium with a capacity for O(10[sup]4[/sup]) people.”
I would say it’s a relatively new notation. It’s been around during my entire career (so at least a couple of decades), but I don’t recall seeing it in older books. As mentioned above, it is somewhat “casual”, so even in modern books it may be harder to find. It’s a staple of shorter-form communication (for instance, on presentation slides).
I just pulled up the most recent twenty new posts on the arXiv under the “particle physics theory” category. Two of those twenty articles have an example of usage [2] in their abstracts.