I have two types in my yard although I have never researched either for a name.
One produces small, bright red berries just like this image and the other produces a dark purple berry.
To me, they are just honeysuckle. I really do not care which. My Robins and my resident Brown Thrasher pounce on the red berries very quickly. The purple berried one is enjoyed immensely by the Catbirds later in the Summer.