His phrasing is terrible. Given that your location is Germany, I'm going to guess that this was a German professor speaking English...which is sort of hit or miss in quality. I'll reexplain what I presume he/she was trying to convey in what will hopefully be a clearer manner.

Let us suppose that we have a patient's DNA in a tube and we'd like to determine their genotype at a certain position. The method of genotyping and actual position isn't relevant for this example (i.e., we'll ignore all of life's complication). At that position, the reference genotype is A. We know nothing about the patient's background or family to alter our background expectations about the genotyping outcome. Consequently, we expect that the patient will simply match the reference (i.e., have an AA genotype). Given that, we have 4 possible outcomes:

                    Observed Genotype
Actual Genotype     AA     AB
AA                  True Negative     False Positive
AB                  False Negative     True Positive

A true negative means that you see no difference from the expected and that this is correct. A true positive means that you've observed a difference and that that is correct. The false positive case is exactly as you described. The false negative case probably only confused you since you took "ground truth" to be "background expectation". In that case, you observed the reference genotype (i.e., the negative state), but it was wrong (i.e., false).

I should note that the "positive" and "negative" terms are more understandable in the context of finding differences. For example, if you were measuring the difference in height between members of two groups and found there to be no real difference, but there actually was (let's assume that the you just didn't measure enough people), then that'd be a false negative result.