The plot thickens. You were sort of on the right track in your previous question, but your description differed from the USB spec and I didn't make the connection of where you were going. There are a number of things that could be your problem.
It isn't clear if your term "auto adapter" refers to a car adapter or one that automatically negotiates the proper power (although they aren't necessarily mutually exclusive). If your adapter has a USB 2.0 port, the fact that it will supply 2.1A won't help your Touchpad. That would be designed to power "dumb", high-current appliances.
USB Power Delivery Spec
There is a special Power Delivery (PD) specification, part of the USB specs. To use more than 7.5 W, the device "negotiates" the power it needs from a "smart" power adapter over a data line. So there is a "data" element to this, but it's not the case where the power pack communicates how much power it's delivering.
There are six predefined power profiles, which define a voltage and a maximum current.
Chart courtesy Wikipedia, which also has an overview of the PD spec.
PD-aware devices interface with the power source through a bidirectional data channel and request access to a certain level of electrical power. It doesn't need to use the full amount of power available from the profile, it uses what it needs within the profile's limit.
So the high current available through the PD spec requires an intelligent power supply and a USB 3.0 or higher connection. Without these, power is limited to the basic USB spec.
Touchpad requirements?
I didn't find the Touchpad manual online, but from various posts, it looks like it requires only 2.0A, rather than 2.1A. If that's correct, it wouldn't necessarily require a proprietary charger.
I did a quick Google search for USB 3.0 car chargers and came across this example, which claims to be compatible with most tablets.
I don't know if this would work with the Touchpad.
I find the 2.4A value shown on the charger interesting. It isn't in compliance with the PD spec. So either it's useless marketing hype, or there is a proprietary spec for current higher than 2.0A broadly shared for tablets. The same thing can be said of the 2.1A rating you describe on your adapter.
BTW, with this charger, if the car voltage is lower than 11.8V, it gives a low voltage warning. So even if the chargers you tried were of this class, using it with the engine off or idling may not have been adequate.
Update
From the comments, the link in your question, and some more research, It looks like the Touchpad predates the current PD spec, and may even have a USB 2.0 connection. This likely means that the charger I posted above, and any similar ones with a USB 3.0 connection, won't work.
Charging at 2A requires some form of handshaking between the Touchpad and power supply. If it isn't the current PD spec, it's something else (older or proprietary). Any power supply will need to be compatible with that. Without that handshaking, the Touchpad may be limited to drawing 500 mA if it's a USB 2.0 connection, or 900 mA if it's USB 3.0.
You may be onto something with the resistors you mentioned in the question. Prior to handshaking based on exchanging data, it was common to identify equipment profiles of various kinds (including things like monitor resolution), by simply shorting, grounding, omitting, or tying a specific voltage to certain contacts in the connector. What your link describes sounds like something HP did along those lines, using resistors to force a specific voltage on the data pins of the USB 2.0 connection to select a high-current charging profile. I haven't yet gotten the USB 2.0 Power Delivery specs, so I don't know if this was part of that standard, or something proprietary to HP. That would mean, though, that only a power supply designed to interpret that will work.
Alternative
Here's another approach--use a power inverter in your car. These convert the car's DC 12V power to 120V AC power. They have limited output power, but you only need a trivial amount. Something like this, for example:
This particular one has a built-in USB charging jack, which would be of no use for the Touchpad. You plug your wall charger into the inverter and that works the same as it does at home.
By 'auto adapter' do you mean it's a car cigarette lighter adapter? Have you tried it with the engine running? The socket normally provides more power (14.5v vs 12v) and the adapter may require the full amount. Also, have you tried a different USB cable? – James P – 2015-12-26T10:18:54.170
@James All excellent questions. I have updated the question above with answers. Thank you for asking. – RockPaperLizard – 2015-12-26T23:00:12.620
OK, a new solution for you guaranteed to work. See updated answer. – fixer1234 – 2015-12-27T01:17:33.310