The black ones are referred to as blackberries as well in English. I've never seen the white ones before. Google translate calls them mulberries, but I've never seen white mulberries before. We had a mulberry tree on our farm as a child, and it gave dark reddish purple berries similar to blackberries.
The only difference (besides flavor) is that blackberries grow on a thorny bush, whereas mulberries are the fruit of a tree.
I guess you could call the 2nd one a white mulberry.
If your first picture grows on full-sized trees, it's probably a mulberry. If it grows on a short thorny bush, around 3-5 ft. tall, then it's a blackberry.
Without seeing the plant, it's difficult to tell, since they look similar.