As I understand it, IV is defined as an annualized 1 standard deviation range for an underlying.
That's incorrect. That definition is for Historical Volatility, which is not the same as Implied Volatility (or IV).
what is the meaning of implied volatility for a strike?
Implied Volatility is one of the parameters of many option pricing models. Simplifying:
call price = f(underlying price, strike, time to expiration, dividends, risk-free rate, implied volatility).
For a given option (say, call) contract, you know
time to expiration by the contract definition. You know
call price by looking at the market. You know
underlying price by looking up information about the underlying stock. You know the
risk-free rate by tracking treasuries rates. The only unknown is
implied volatility. So you can solve the equation and find the IV.
That is the definition of IV.
- how do you get a value for the underlying as a whole?
There's no standard definition of "the IV of the underlying", nor "the IV of the underlying for month X". You'd have to ask your broker to know exactly which definition they are using. I don't know what Tastyworks use.
It's typical to follow the definition of the VIX, so you can look that up on the Cboe website.
- Does this mean that someone buying the 250 put for the current market price is valuing the volatility for this cycle at 23.76%
Not exactly - like mentioned before, there's no such thing as "the volatility for this cycle"; at least not as a standard definition. What's happening is that the market is pricing that 250 Put at 23.76% implied volatility -- in other words, 23.76% is the "magic number" that is found by solving the equation defined by the option pricing model you are using.
The difference in IVs that you mention is very common in the marketplace - it's commonly referred to as volatility smile. There's no single explanation for the phenomenon.
One common explanation is the fact that market returns are not normally distributed - there are fat tails - while option pricing models typically assume normal or log-normal returns.
The fat tails mean that prices jump really far more often than predicted by normal distributions. If the IV of OTM options was the same as ATM options, those OTM options would be underpriced - that is, the buyer of those options would have a positive expected return. But the marketplace prices contracts in a fair way, otherwise arbitrage opportunities would exist. That's why IV is higher for OTM contracts.