When you create the network dataset, what connectivity model did you choose: End Point or Any Vertex?
If it is End Point, then you will not get the node to connect to a middle of an edge as shown in your picture.
In that case, switch to Any Vertex and re-build your network.
Thank you for your explanation, it definitely helped me understand the situation a bit more. However, I still do not understand why I have to use the Integrate tool to my features before generating my network dataset. Even if I set a higher tolerace (like 0.002) to my original dataset and then generate the network dataset, there are no connections. Even if the nodes are only 0.001 m away from the edges. Shouldn't it work without using the integrate tool?