My Danish cousin is the one looking in the Danish records for Magda, as it is a bit easier for her being Danish. Here is what she says:
Yes, she is not with her family in 1911, I searched the whole of Rantzausminde/Egense and she is not there. It is of course true, that there is a slight possibilty that who ever did the census all those years ago, made a mistake. But how could we ever know! I have to assume she was somewhere else and that hopefully I will find her.
Magda moved around quite a bit after coming to the US. After having Paul in New Jersey in 1916, the one we are assuming was fathered in Denmark, she had a daughter, Elsa in 1918 in Montana, we find her in North Dakota living with a Danish man , listed as a servant, with her two children, Paul and Elsa, then in 1921, she has another boy in Minnesota, she is alone then. She then marries my great grandfather, she is actually ~2 months pregnant with the 3rd illegitimate son at the time......
So its safe to assume she probably didn't know she was pregnant when she left Denmark and that she left because she wanted to, at least somewhat, and probably as a result of the affair scandal in Svendborg.