While Bid/Ask is $13.5/15.5 the last actual transaction was done for a mere $12.15 = instead of higher a lot lower.

It's a good rule of thumb always to ignore the last trade price of any option. It is usually anywhere from hours to days old and gives no current information. Look at the midpoint of the bid/ask instead. In this case, the "price" of the option was $14.50. You'd probably pay 50 cents more if buying and get 50 cents less if selling, but the midpoint of the current national best bid and best offer (NBBO) is the "official" price when brokers are doing statements at end of day, for example.

Jim repeatedly said options move strongly when their strike price is reached.

To break it down more...

The time value of the option price is at its highest when the stock price is at the strike price. The time value alone is a type of bell curve shape.
The total option price is the sum of the time value (that bell curve) plus the in-the-money or intrinsic value, which is a wedge which hits zero at the strike price.

The price of the option doesn't move most quickly when the stock price is at the strike. The amount (in dollar terms, not percent terms) that an option price moves with each $1 move in the underlying stock price is called its "delta". This is actually highest for a deep in the money option, when it approaches 1. The option price becomes asymptotic to its in-the-money value.

It sounds like you are looking at the variation in the percentage price change in the price of an option, not the usual way to think of it. Viewed that way, yes, an out-of-the-money call option will start to rise in price more quickly as the stock price rises up through the strike, because the price move suddenly starts to include intrinsic value (which moves $1 for each $1 rise in the stock price), not just the slowly changing time value which was its only value when it was out of the money. Honestly I'm not sure what the exact function is, but I think it's close to this: "for each unit rise in the price of the underlying, an out-of-the-money call option rises in percentage terms a relatively constant amount, but as it goes into and further into the money, for each unit price rise of the underlying the call option rises as smaller and smaller percentage amount, the percentage falling only slowly".

For puts, just reverse everything : )
Plus the little "extra" when you remember that puts are much more in demand on panicky market days, so their prices are often momentarily higher. Like a hard insurance market.

This all ignores the variation caused by the falling amount of time till the option expires.

Jim