What you are actually trying to change
You want to change the icon displayed for an application in the Dock, because two applications have similar icons and you want to make one visually distinctive.
Apple's current documentation about Get Info → small icon → Paste is specifically a general procedure for changing the icon of a file or folder.
It does not provide a separate procedure for changing the Dock representation of an application. (Apple Support)
Apple's developer documentation does establish that an application's icon is associated with its Dock tile: NSApplication has an applicationIconImage property, described by Apple as “the image used for the app's icon,” and Apple specifically discusses it in connection with the Dock app tile. (Apple Developer)
So you were right to point out that Apple's page I gave you wasn't really addressing your specific objective.
I also shouldn't assume that the Get Info procedure will or will not work for your particular application without knowing which application it is.