I think they refer to market value or in this case market capitalisation, which is well defined as share price * # of shares. Both prices you listed are just the asking prices, the true market value will be somewhere between the bid and ask prices. The bid ask spread in a liquid market is small under 'normal' circumstances, eg Apple shares bid/ask is currently 243.50 x 400/244.00 x 2600, people willing to buy 400 shares at $243.50 and sell 2600 shares at $244.
I'm not sure how the asking price for a share is substantially different than the concept of "value" I mentioned before. Both are fictitious concepts that have very little to do with the actual price something should have (I'll use the term "worth" here to describe that more precise measure, but feel free to substitute average cost or some other term that means "the objectively calculable price for something").
A share price can be both under and over valued right? Under or over what? Worth.
One way to measure is revenue/outstanding shares (though secondary factors like P/E, private investments, etc. figure into this to define more precise measures) another way is based on a softer metric like "valuation" which is typically some multiplier of revenue/yr (notice the root word of valuation is "value" which is precisely what I'm talking about -- and we both know that valuations can have very little to do with reality (see Facebook for a case study in how valuations can be broken).
Share prices are merely a representation of the subjective "value" of a share vs. the actual price for some % of a company. If the market thinks something is hot, it doesn't matter if a share is actually worth only $2 or $3 from a pure revenue/yr perspective, demand for those shares can drive shares up to ridiculous levels -- hundreds of percent above what they should be. If a company is not "hot" (based on the very subjective moods of the market) those shares might even sell for less than they are worth because people see the value as less.
The market does a very efficient job of obscuring the worth of a share in some company by only feeding investors value signals.
There is nothing different from this concept than the concept of value at the microeconomic level. If I see an item for sale. I have a pretty good idea what it's worth is based on materials, construction techniques and various other factors that contribute to the manufacturing cost of a product. But again, just like in the market, all of the signals I receive about a product are value signals not worth signals. So I compare that mental model to the asking price to determine if I'm getting a good value for my purchase. Value here being how close to the average per unit cost I'm able to estimate. A pair of shoes might be priced to sell at $150. The next week it goes on sale for $100. That increases the perceived value I get from the purchase. But I might still skip it if I have a good idea that the average cost per unit-pair of shoes is only $10. But companies seek to increase the value of their products at higher prices by various means like brand building.
This is the logic that good investors use when deciding to buy and hold or sell shares. If a stock's value is perceived to be too high compared to the average worth of a share, investors expect that the share price will more likely go down than up. Companies seek to influence this by keeping investors interested with positive statements about the company. "new product!", "big contract!", "how new business concept!", "change in leadership!", "increase in business efficiency!" so that a share of XYZ company, that may only be worth $10 on a revenue/shares matter is valued at $150 by the market.
Of course what you say is true, value or worth is subjective - something that you think is worth only the cost of production may be worth a lot more to someone else. Everybody has their own value metric, but funnily if you want to buy or sell at a particularly instance in time you have to pay the market price.
