Hi @wrongwright ,
I can’t explain it like you’re a 6 year old, sorry (I don’t know enough abou EEVEE), but a Mesh light is a Mesh (a bunch of points in space (Vertices), connected by Edges and Faces), with an Emiisive Material (Emission Shader, for instance). An HDRI is connected via a Background Shader and that is also an Emission Shader. Emission Shaders do not light other Objects (and thus casts no shadows) in EEVEE, but are just glowing a bit. But they can affect Light Probes and cast shadows via indirect light (I don’t know how that works, sorry).
A Lamp (nowadays called a Light in Blender), is one of four types; Area, Point, Spot or Sun and does light other Objects in your Scene (and therefor casts shadows) in EEVEE.
Everything that is said here is about EEVEE, lighting with Cycles is covered in last Chapter.
