+1 for running a NUC on 12V. Just watch your power cable size - the peak current goes from like 3A at 19V to 5A at 12V. The coax power cable from the stock PSU won't cut it.
As for choosing a PC, start with what you need, and work from there. Number of ports and types, storage (250 GB is more than enough). 8-16GB of RAM, with 8 as a minimum these days. Touch displays still add a premium to the price, but that's a matter of personal preference.
Form factor is worth thinking about - laptops and convertibles win for portability, but they are mostly a fixed configuration. A Real computer with external display, keyboard, and mouse gives you more flexibility in where you put things. You can also defer purchase of a bigger display to later, which helps with cash flow :-) There are some nice 'small form factor' PC's out there, so you don't need a huge tower chassis.
Once you have requirements, it's a lot easier to shop and compare. Look for sales - the US President's Day sales are mostly gone, but there will be more.