What is the value of providing context-specific icons over a generic icon in a dialog?

Background: Need to design icons for Dialogs in a medical enterprise application. These icons(non-actionable) will be used in Dialogs like Error, Warning, Alarm etc.

Question: What is the value of providing context-specific icons over a generic icon for a dialog? For eg. If an Error message is thrown at the user while performing a Task. Let's assume that the error was caused due to a network disconnect. Would it help to represent that Error message with a generic error icon or the icon need to be specific to the nature of the problem Ie. an icon representing network + error vs a generic Error icon

Larger concern here is from an experience standpoint for a user consuming these dialogs. Will the user understand and respond better to dialogs if the icons were context specific and be more aware of the nature of a dialog.

Need some sound UX advice on this.