E6540 with just two external monitors (non-duplicated display)?

@deladatu  Have you tried connecting one external display, then setting your built-in display to "Disconnect this display" and THEN connecting the second external display?  The GPU in that system might only support two displays total.  Otherwise, there are some Latitude systems that have two display outputs sharing a common GPU output, which means only one can be used at a time, but thus far I've only seen that implemented on systems that have 3 outputs, such as USB-C, HDMI, and VGA.  I'd admittedly be surprised if that design was implemented here.  That said, displays should be plug and play, and if you can get both displays working separately but not at the same time even after disabling the built-in display, that might be what's going on.

One option you could consider would be to get an E-Port Plus docking station.  They're not very expensive anymore.  You'd also need a 130W power supply if the dock you bought didn't come with one (some sellers now sell just the dock unit), but those aren't expensive either.  That would give you two DisplayPort outputs and two DVI outputs, and you could use any combination of two of those simultaneously.  You could use DisplayPort to HDMI/VGA adapters if needed.  If you decide to consider this, also be aware that in addition to the difference between E-Port and E-Port Plus (the latter has two of each type of display output), there are also two generations of the E-Port dock models.  The first generation only has USB 2.0 ports.  On the second generation, the two rear USB ports are upgraded to USB 3.0 (though the ones on the left edge are still 2.0), and the DisplayPort outputs were upgraded to DisplayPort 1.2 to support higher end display setups.  If you care about these differences, the way to tell the generations apart is that the rear USB ports are blue on the second generation (indicating USB 3.0 support) while they're black on the first generation.

@deladatu  How did you establish that one port is coming from the Intel GPU and one is coming from the AMD GPU?  And which is which?  Most dual GPU laptops have all outputs wired to the Intel GPU, and the laptops that have other designs typically have them because they're designed for use cases where that can be beneficial, which is typically laptops meant for gaming and/or high end workstation uses.

But the reason I specifically asked about setting the built-in display to disabled before connecting the second display was because if the Intel GPU does in fact have control of all of the built-in display output connectors, then if it also only supports 2 total displays, you might have to disable the built-in display to free up a GPU "slot" before you can light up a second external display.  Otherwise, the system might assume that you want to keep the built-in display active, in which case yes you'll only be able to light up one additional display.  I've never used that exact system, but given that it can run dual external displays while docked, I would be quite surprised if it couldn't do the same thing using its other built-in display outputs even after setting the built-in display to disabled.  But it's an easy test.  Connect a display to the HDMI and VGA outputs separately to confirm both outputs work.  Then connect only one display, go to Windows Display Settings, set the built-in display to "Disconnect this display", and then connect the second external display.  If it doesn't work, I guess that's how it is.  But it's not clear from your reply that you've tested that setup yet.

> How did you establish that one port is coming from the Intel GPU and one is coming from the AMD GPU?

I think I misstated... I don't remember exactly which resource I found the specifications on for the laptop's video out ports, but I think I determined that it will only support 1 external display from those built-in ports.

I may be able to use one of the USB 3 ports as a 2nd external monitor, but I haven't researched that far yet.

I do have the dock which supports 2 external displays. However, when the laptop isn't in the dock, it would be nice to have 2 external displays directly out from the laptop, but I'm not seeing that it's possible yet. It appears that it only supports its own display then an external one, which is probably fine for the non-dock situations.

@deladatu  There are USB dongles that can be used for displays, but the ones that plug into USB-A “regular USB” ports (rather than USB-C, which your system doesn’t have) work by using “indirect display” technology such as DisplayLink — not to be confused with DisplayPort.  These dongles can indeed be useful for running more displays than your system has ports or GPU support for running, since displays connected via DisplayLink don’t require a native GPU output or count toward the GPU’s maximum number of simultaneous displays. However, the way DisplayLink works comes with drawbacks that can be significant in some use cases. I’ve written about those in the post marked as the answer in this thread. But if you want to explore that route, Plugable makes DisplayLink dongles that give you either an HDMI or DisplayPort output, and they have a solid reputation for quality products and knowledgeable support reps if needed, so I’d look into their products.