I don't really know how to describe this problem, but I just want a clean cut if you know what I mean. It seems like the ellipse is trying to form an ellipse again and so there are new lines in the middle, how can I get rid of the lines in the middle? Adobe Illustrator does the same thing.
What you’re after is possible. There’s two approaches I can think of that work.
If you are happy with a destructive approach, expanding the stoke prior to the boolean operation will give you want you’re after. To do this select the ellipse layer and choose
If you’d like to maintain the ability to edit the stroke width, you’ll likely have to use some of Affinity Designer’s masking abilities. I’ve masked the circle with a compound path of two rectangles.
To do this, drag the layer you’d like to be masked to the layer you’d like to mask it with. Drag to the area just to the right of the layer thumbnail.
Given it’s for a logo, using a destructive method is probably best for the final assets.
Break the residues of the ellipse at the corners. Then delete the straight parts. You have the "Break Curve" tool in the context menu when you use the "Node" tool.
This leaves the ends as usual open curve ends. Should they be along the cutting line (=horizontal) - then this method does not work.