There's currently no way to format a "Display Text" effect so that it looks like NPC speech, to display speech to the player that doesn't intrude on other players.

After thinking about it, there could be new options in the "Display Text" dropdown to format the text as an NPC speech, with correct chat coloring.
Another option would be to add color tags to the Display text effect. It means colors wouldn't adapt to each player's settings, but they would have more freedom in the choice of colors.