The information on the UK government web site you linked seems clear to me. I suspect you have simply misread it.
The relevant part appears to be:
Customs relief for non-EU vehicles
You may be able to claim relief from customs charges if your vehicle is registered outside the EU and you’re bringing it to the UK temporarily. This is known as ‘temporary admission’.
Fill in form C110 and take your vehicle through the ‘nothing to declare’ channel when you arrive in the UK.
This doesn't apply to "non-EU residents", it applies to vehicles registered outside the EU (and Texas is quite far outside the EU).
Also you must make sure that the vehicle's inspection and registration sticker is current before you go.
You must not be bringing the vehicle to the UK in order to sell the vehicle there or to establish residence in the UK and must export it again in order to qualify for the customs relief. And as the page also mentions, you will need the customs relief form if you are stopped by police.