The distinction matters, though. In a system with surprise rounds, an extra action is granted to "those who have surprise." In the 5E system, actions are denied to "those who are surprised." The key difference is what happens to those who neither have surprise nor are surprised--in this example, the visible bandits standing in front of the PCs.
Under the surprise round system, the visible bandits do not have surprise, so they don't get to act on the surprise round. The snipers are the only ones who can act. They take their shots, and then combat proceeds.
Under the 5E system, everyone can act on the first round except those who are surprised. So the visible bandits would get to attack just like their sniper companions. If the PCs are surprised, they will be the only ones not acting.
This is why I argue that there should be no surprise in this encounter. The PCs know there is a threat, even if they don't know its full extent. They are already braced and ready for trouble. Surprise is when trouble hits somebody who isn't expecting it. The proper way to handle this scenario is for everyone to roll initiative, and then start combat on the snipers' initiative. (Effectively, everyone who rolled higher than the snipers used their action to talk.)