Do diodes care about current? or only voltage?
Yes it matters. Despite how we think of diodes the voltage across it is dependent on the current passing through it. 

Diodes are actually non-linear resistors. How much resistance they present is dependent on the voltage across them. But like a resistor, how much voltage across it is dependent on how much current you feed through it. (V=IR) 

The conductivity (1/R) of a diode is depicted with the non-linear curve shown below. The zener action is on the left side of the chart.



The Zener voltage given on the spec sheet is at a specific current. Less current = less voltage, more current = more voltage. 

Fortunately, once past the knee of the breakdown curve you need to change the current a lot for it to make a lot of difference in the voltage. However, you have to get it to that point first with sufficiently nominal bias current.

Below is a typical zener diode spec sheet. Note it indicates the tested current for the specified voltage. 



Look at the Zz and Zzk figures. These are the resistances of the diode past and under the zener knee point. With this diode in order to get 5.6V across it you need at least 3.5mA. (5.6/1,600)

When designing your circuit you should nominally match the circuit to the chosen devices indicated current. When choosing a zener, select one that more closely conforms to your "current range".

NOTE: The diode listed above would not clip at 5.6V for the 10K resistor shown in the OP's question. The zener will just act as a 1.6K resistor when reverse biased. You end up with a resistor divider and a much smaller signal than you expect.

Of course too much current and you will cook the diode too.