Context of the question:
Schlosshauer (978-3-540-35773-4, p. 33) states:
"A useful intuitive way of quantifying the entanglement present in this state [(1)] is to consider the following question: How much can the observer learn about one system by measuring the other system?"
Taken at face-value, the phrase above doesn't elaborate on what measurement (in what basis) is meant. So one could assume that entanglement strength of state (1) is a property that is defined "relative to two certain measurement bases" - one for measurement on the first and one for measurement on the second system.
Although I have the suspicion that that's not the case: Even if (1) is a product state one could learn "a lot" about system 1 when measuring system 2 for a certain pair of measurements (measurement bases). The difference seems to come down to the used measurement bases or rather - whether correlations appear in any measurement basis. Therefore, there seems to be some clarification regarding the measurement bases missing in Schlosshauers characterization.
I tried to improve the cited phrase.
A useful intuitive way of quantifying the entanglement present in this state [(1)] is to consider the following question: How much can the observer learn about one system by measuring the other system (regarding measurements in any bases)?
Unfortunately, this is not satisfying either since there is a pair of measurements where there is no correlation, even for the entangled state - namely if $|\psi\rangle=|00\rangle+|11\rangle$ and one measures $O=|+\rangle\langle +|-|-\rangle\langle -|$ on both systems.
So, a second try:
"A useful intuitive way of quantifying the entanglement present in this state [(1)] is to consider the following question: How much does a measurement on one system change measurement behaviour of the 2nd system?"
Question: Is entanglement of a bipartite system (if defined by correlation between measurements) a property that is defined relative to certain measurement bases? If not, then Schlosshauers explanation is somewhat misleading, since it doesn't mention measurement bases at all. How would a correction of said explanation look like then?
It seems to me that Schlosshauers description is correct - I was able to find it similar in the accepted answer to this question. Therefore, I am probably misunderstanding the phrase "How much can the observer learn about one system by measuring the other system?".
So far I took the view that it must be understood this way: If there is e.g. a measurement on the product state $|0\rangle_A\otimes|0\rangle_B$ that first measures in the $|0\rangle,|1\rangle$ basis on system A, you would always get the result "$|0\rangle$". If one measured system B after (in the same basis), you would always get the result $|0\rangle$". In this sense, you could learn something about system B by measuring A. But that is not how said phrase is meant. It is meant in this way: Measurement on system A influences measurement on system B. In other words: Measurement on system A changes the measurements statistics of a subsequent measurement on system B in comparison to solely measuring B (without measuring A first).
If somebody could doublecheck what I wrote in my update, I think my question has been solved! Thank you.