complexity theory NP Ok, I really need help because I have read in so many books but still don't understand the complexity class NP.

These are the books:


Theoretische Informatik; Katrin Erk, Lutz Priese (german)
Grundkurs: Theoretische Informatik; Gottfried Vossen, Kurt-Ulrich Witt (german)
Theoretische Informatik; Ingo Wegener (german)


Links: 


https://de.wikipedia.org/wiki/NP_(Komplexitätsklasse) (german)
https://www.ibm.com/developerworks/community/blogs/jfp/entry/no_the_tsp_isn_t_np_complete?lang=en (english)


And I actually have more links but I can't post them here because I don't have enough reputation 

So far I know: 


A problem L is in NP when there exists a nondeterministic algorithm which solves problem L in polynomial time.
NP includes decision problems. If there exists a solution for these problems, it should be possible to check with a deterministic algorithm in polynomial time whether the solution is indeed a solution.


To 1.: is the calculated solution an exact match or is it just an approximation? To 2.: is it just necessary to be able to check in polynomial time? For example: the solution is not indeed a solution but I was able to check it in polynomial time. Is the problem still in NP? 

For a problem to be in NP, I must have a nondeterministic algorithm which solves the problem in polynomial time. The solution is an exact match and I must be able to check in polynomial time with an deterministic algorithm that the solution is indeed a solution???????????????? 

Can someone please help me to understand. 

Further thoughts: 
Let's take the TSP as an example. 
Question: Does a tour exist &lt;= k ?

Let's assume I wrote a nondeterministic algorithm, which calculates a tour &lt;= k. So my answer would be: Yes, a tour like that exists. 

However Vladimir wrote in his answer: If this question can be answered by a polynomial time nondeterministic algorithm, then the language L is in NP. 

So I gave an answer 'yes'; that means the TSP is in NP?! However what about if my answer would be: No, a tour like that does not exist! ? Then the TSP would not be in NP? 

And looking on the other definition: Further I answered: Yes, a tour like that exists, because my nondeterministic algorithm calculated a tour like that. Now if I can check with a deterministic algorithm in polynomial time that the calculated tour is indeed verified to be correct, then TSP is in NP !?
First of all note that a non-deterministic algorithm and a non-deterministic Turing machine are two different things.

A non-deterministic Turing machine is a computational model, while a non-deterministic algorithm "is an algorithm that, even for the same input, can exhibit different behaviors on different runs, as opposed to a deterministic algorithm" (Wikipedia). For example a QuickSort algorithm which partitions the array based on random choice of a pivot is a non-deterministic (probabilistic) algorithm. 

So your definition 1 should be read as following 


  A problem $L$ is in $NP$ when there exists a nondeterministic Turing machine which solves problem $L$ in polynomial time.


Similarly, the second definition should be read as following:


  NP includes decision problems. If there exists a solution for these problems, it should be possible to check on a deterministic Turing machine in polynomial time whether the solution is indeed a solution.


For the concept of $NP$ see this post.