While I know there is an option in the settings to remove the Lumber Mill from AoP if you have Lignumis I realized there is one difference that can make AoP's lumber Mill useful and something that can be done to differentiate them. Lignumis's Lumber Mill is a burner machine while AoP uses Electricity.
If you call the AoP Lumber mill an "Electric Lumber Mill" and add either a lightning bolt icon to it and/or give it some kind of tint or accent coloring to make it look like an upgrade would be a nice solution (though I'd still suggest keeping the setting to remove it)