Suppose I pay $60k in cash and take out a $240k mortgage to buy a house. The $60k is equity (market cap), and the $240k is debt. The value of the home is $300k. It works the same way with companies.
Now suppose someone wants to buy the business. I still have $60k worth of equity (100%). Do you think they can buy the entire business (claims on ALL of it's cash flow) for $60k? Of course not. They'd have to pay off the debt AND buy the equity. That number is $300k - the enterprise value.
Market cap is defined as the value of the outstanding equity (share price x number of shares). Period. The value of the company can be, and often is, much greater because most companies choose debt as part of their capital structure.
Startup tech companies like Instagram rarely do that. Nobody lends us money, so we're stuck with equity. So for Instagram, Market Cap (if there is such a beast for private companies) IS the enterprise value. For the NY Times, market cap is but a percentage of the enterprise value. So it turns out the headline of this article is actually quite incorrect. The NY Times, in any way you can measure, is clearly worth more than $1B.
So in your latest example, and also throughout the thread, you're confusing (or using interchangeably) the concrete sale price and a fuzzy intrinsic value. Why would the buyer of your business pay $300K upfront? All other things being equal, the sale price of the business would still be $60k, and they can be paying off debt for the next 15 years for all I know. The conventional media-reported value of the business would still be $60K, and it won't be very misleading.
In addition, not all debt is created equal. A company like Google can probably easily get, say 1B of super cheap debt on favorable terms, while something like Barnes & Noble would barely get the same 1B at much higher interest rates and repayment schedules. It's still 1B in enterprise value, but it's clearly not the same debt.
You also mentioned cash flows. So to continue with your example, if your business is actually losing money hand over fist, and has negative cash flow on top of $240K of debt, what do you think its value would be? Would it be equal to your simple enterprise value formula? What if the same business is growing at 100% per quarter with super high operating margins? Enterprise value can be just as misleading.
You'd still owe $300K in mortgage, and be out $200K of cash, but that doesn't mean the house is worth $500K.
P.S. And what's up with downvoting?