The direction reverses between calls and puts, and this is the single commonest error in the whole subject. A call gives the right to buy at the strike, so it is worth using when the strike is the cheaper of the two prices, which means below the market. A put gives the right to sell at the strike, so it is worth using when the strike is the dearer of the two, which means above the market. Stated as a rule to remember rather than memorize: an option is in the money when exercising it would beat transacting in the open market. The SEC's two sentences are that rule applied twice.
Only an in-the-money option has intrinsic value, and everything else in the premium is the price of time and expected movement. Intrinsic value is what the right is worth if exercised immediately, so it is zero for an out-of-the-money contract and zero for an at-the-money one. The rest of what a buyer pays reflects how long the contract has to run and how far the underlying is expected to move. The SEC's own list of what determines a premium is exactly those three inputs: "the underlying stock price in relation to the strike price," "the length of time until the option contract expires," and "the price volatility of the underlying stock." The first of those is this page's subject; the third is covered on our page for implied volatility.
A deep out-of-the-money option is cheap for a reason that is easy to misread. Its whole premium is the second component, so it is a bet that the underlying moves far enough, in the right direction, before a date. That combination of conditions is why the contract is cheap, and price alone tells a buyer nothing about whether it is good value; a low premium on a contract that needs an improbable move is not a discount.
The margin rules formalize the same measurement, which is a useful confirmation that it is a real quantity rather than a manner of speaking. FINRA Rule 4210(f)(2) uses "in-the-money amount" and "out-of-the-money amount" as computed figures, and defines the latter for stock options as, for a call, "any excess of the aggregate exercise price of the option over the current market value of the equivalent number of shares of the underlying security," and for a put, "any excess of the current market value of the equivalent number of shares of the underlying security over the aggregate exercise price." Margin on a short option may be reduced by that amount, subject to floors. So the distance by which an option is out of the money has a dollar value in the rulebook, not only in conversation.
Being in the money and being profitable are different questions, and conflating them is how a holder concludes a losing trade is a winner. Crossing the strike changes the option's status; it does not repay the premium already spent. A call buyer is ahead only above the strike plus the per-share premium, and a put buyer only below the strike minus it. Our pages on the call option and the put option work each of those through with numbers.
Where the state actually decides something on its own is at expiration. Expiring standardized equity options are subject to the clearing corporation's Exercise-by-Exception procedure, under which contracts in the money by specified amounts are exercised automatically unless the holder gives contrary instructions. So the three states are not merely descriptive on the final day: which one a contract lands in determines what happens to it by default. Our page on options expiration covers that machinery, and our page on options assignment covers what it does to the writer.