What is being tested
Your (1) indicates that you tested for 0 difference in population medians and your (2) indicates that you tested for the population median of pair-differences being 0.
Strictly, a signed rank test is not testing what either of your interpretations say.
The actual population quantity being considered is the pseudomedian of population pair-differences (across all population pairs); the test relates to the sample pseudomedian of pair-differences (1-sample Hodges-Lehmann estimator), and the corresponding estimate is based on it.
If you have symmetry of pair-differences that will be the same as the second interpretation, but the test only needs that symmetry under the null; it can be perfectly easy to interpret in a wide class of cases when you don't have symmetry under the alternative.
[For example consider pairs of positive observations with no change in distribution under the null and a scale shift under the alternative. The test is perfectly suitable under both the null and alternative but in general under the alternative both your interpretations of what is being tested are going to be wrong.]
Consequently, you cannot easily address the suitability of this assumption from observing the data because you don't know whether the null is false*.
However, if the symmetry assumption does hold under the null (and the considerations here can sometimes be addressed by a simple argument of no effect), then the (under the null) the population pseudomedian of pair-differences will correspond to the population median of pair-differences.
A rejection doesn't automatically imply that the same applies under the alternative, however (at least not that I can see).
If you can make an argument for symmetry under the alternative, it would still correspond to the second interpretation, but - while it's often fairly easy to make the argument under the null - it's much more difficult under the alternative.
It may be safer to stick with what the test actually does look at.
* for that matter you shouldn't be choosing a test based on what you find in the data
Hypothesis tests are about populations
Generally, hypotheses are about populations*. That's usually the point of them, to make some kind of inference about some population that you cannot wholly access. The word 'significant' doesn't belong in the hypothesis nor in a conclusion about it.
* There are some situations (with randomization to treatment group) that you can perform a form of hypothesis test that doesn't require a random sample of some population to make a conclusion about the effect of the treatment on the units used, but typically people want to make some kind of inference beyond the sample. (e.g. It's not necessarily much use figuring out that a treatment applied to middle-aged male professors at a large university has a non-null effect if you want to be able to use it to treat non-middle-aged non-male non-professors not from that university.)