The second quote sounds like complete nonsense to me. There is no "high cost to their owner" from converting the stocks into cash. The high cost comes from the stock losing value. That an asset can lose value does not affect whether or not it's a liquid asset.
As the first quotation explains, an asset is liquid if it can readily be converted to cash with little impact on its value. Even if you sell a stock at a loss, the selling process itself can be readily done and has little impact on the value (which is, of course lower) of the asset.
I think they're trying to make the point that if you want to treat a volatile asset as liquid, you run the risk that when you need cash, you'll have to sell the asset at a time you'd rather not sell it. But that's true of any liquid asset and even of illiquid assets.
But their thinking is fundamentally and seriously wrong. A stock that has gone down is not therefore more likely to go up. And even if it was, that you held the stock and it lost value doesn't give you any special ability to participate in its future increase.
If you hold a stock and its value went down, you lost that value, period. That is a loss you will always have. If you continue to hold the stock and it goes up in value, you will get a gain. But that's the same gain that anyone who bought the stock after it went down would make. There is no special loss of opportunity from being forced to sell a stock at some particular time. You simply lose the ability to hold that stock, just as you would having to sell a stock at any time.
So the bit about "a need for cash arises when they have sunk to a low price" is meaningless. Any time the need for cash arises, you will lose both the risk of taking losses on the stock and the benefit of further gains on the stock. After the stock has sunk to a low price, it might well continue to sink to an even lower price.
So, again, that second quote is nonsense.