Yes, most English speakers do call both foods "pepper". There are few ways to distinguish them if you need to.
This is also known as black pepper (or red pepper, depending on the color):
On the left, you can see some ground pepper, while on the right there are some pepper kernels. You can refer to the small individual balls as pepper kernels or peppercorns.
The second kind you posted are also known as bell peppers, because they are shaped like bells.
You can distinguish these by color. (E.g. red bell pepper, green bell pepper)
Note that these are also peppers.