OK, the solution is as simple as I am clumsy:

whithin the IvyGen (Thomas Luft, yes) there is a button called "birth".

Following my queer kind of logic I double clicked on my model for the starting point of the growth, then I clicked "birth" (seemed only natural to me), then "grow", and exported the grown Ivy as obj.

Had been my thought - BUT: you've got to press "birth" NOT before growing, you have to click on it PRIOR TO EXPORTING!

Only then it works like it should, and for the whole "problem" >> entirely my bad.

Don't feel actually bad, though :), it works, and I will turn again more to reading handbooks and/or FAQ's.

Wonder what I might have missed with Blender and the remaining greyed out obj's in Vue...

Anyway, thank you for bothering!