Isn't it contradictory the definition of significance level with a significant test result? Definition of Significance level: The significance level is the probability of making a Type I error. I.e. rejecting the null hypothesis given that it is true. Hence, a small significance level is desirable
However, when we make a hypothesis test and can reject the null hypothesis we say the test is significant.
Don't you think there is a contradiction there?

"Rejecting the null hypothesis" is in practice* "accepting the alternative hypothesis".
This is where your confusion might come from.
The term 'reject' sounds like a negative. But actually it is positive from the point of view of the statistical accuracy of an experiment or observation. It meant that we made an precise observation. The measurements were accurate enough to be able to do the rejection. If you have a noisy signal then you could not reject the null hypothesis because of the noise making it impossible to reliably measure an effect.
Also 'reject' has often a positive touch because often the experiment is not about 'rejecting' the null hypothesis but instead about "confirming' or measuring some effect. Significant means that the effect was able to be measured with sufficient precision such that we can differentiate it from the situation without the effect present (the null, empty, hypothesis). From a Popperian viewpoint the scientific knowledge advances by eliminating possibilities; the more we can reject the better.
A related confusion is testing negative on a medical test. Negative is good because it means not sick.

*Important note, the above is more subtle and there is not really literal accepting
  - Expression of significance: or 'acceptance' means that we observed an effect, and consider it as a 'significant' effect. There is no literal 'acceptance' of some theory/hypothesis here. There is just the consideration that we found that the data shows there is some effect and it is significantly different from a case when to there would be zero effect. Whether this means that the alternative theory should be accepted, that is not explicitly stated and should also not be assumed implicitly. The alternative hypothesis (related to the effect) works for the present data, but that is different from being accepted, (it just has not been rejected yet).
