Thank you Ceciliabr! That's indeed made the difference. I'm new to Blender and did not now anything about UV unwrapping.

For anyone reading this who is also limited to Blender 2.79 (because of an older computer): I found these two videos helpful -- details of the interface are different than in v2.8* :

https://www.youtube.com/watch?v=vJOXAQPLYSQ
https://www.youtube.com/watch?v=scPSP_U858k