This is the sense in which we mean value. The next nexus of confusion seems to stem from adding the debt and subtracting the cash. One thing that continues to confuse me is that my house example tends to obfuscate the issue more than clarify it. People are accustomed to thinking of their equity in a house as a separate thing outside of the total value of a house. However, your equity in a house is completely contingent on the current value of the whole house.
Companies are the exact same way.
A company is worth a certain price based on the cash it generates and the assets it has at its disposal. However, from the perspective of someone analyzing the business, the intrinsic value does not change at all. Another reader asked why cash did not increase the value. The reason is simple. The market sets a price on the equity.