Buying a "blender" that can purée a potato
Your requirements appear to be for a "raw food" blender, the motor on those units are typically well above 500 watts (1000+ Watts).  This enables harder/tough foods (such as ice, kale, nuts) to be broken down smooth enough that no sieving is required, with less risk of overheating the motor.  Note: when a blender has a nut bag, or a nut bag option, it alludes to potentially not able to break down harder food without leaving noticeable particles.

There are a few blenders suitable for raw food enthusiasts, depending on each model they may have a:


Tamper
Higher rpm (20,000+ rpm)
Higher power (500 Watts — 1000+ Watts/2–3 horsepower)
Compared to a standard blender (typically 200–400 Watts)
Different jug design/dimensions (width/height)
Different speeds and/or cycles/programmed settings (e.g. smoothie, 60 seconds, 90 seconds, clean, etc.)
Different Blade designs
Smaller portion containers (capsule/bullet shaped)


A tamper is usually provided to allow the user to push food items back towards the blade.  When a tamper is not included on some of the other blenders it may be due to a different blade design along with a different power motor.  This may cause all of the food in the blender to be pulled back to the blade by the vortex it creates, thus alleviating the need for human intervention.

The price variance between blenders is wide, and quite expensive for some of the higher performance models.  Intrinsically some research into refurbished models.



In direct answer to the questions from OP.


  What is it about a blender that allows it to make a good purée rather than just mixing things easily? 
  Does anybody know what makes a machine capable of making a good purée as opposed to just blending, especially when both kinds are sold as “blenders”.


Mainly power of the motor, and the blade design enables the blender to breakdown hard foods completely.  Regular blenders do not have the power, nor the cooling ability for the motor to operate at long enough periods - even some models that market suitability for crushing ice can be known to overheat/fail.


  Should I be looking for ones with different options for blades like a food processor, to make it well suited to different purposes? E.g. I have seen blades just for milkshakes.
  In general, how do you shop for one when the only information a seems to be total power and the number of presets?


Blenders that have a high power motor, high rpm, and/or market its suitability for ice/nut butters/hard spices (e.g. nutmeg), indicate potential.  Where possible, check reviews (where blending nuts has been tested) or better still, when the blender has demonstrations.  You may be able to request a demo on nuts, if so you could taste or strain this into a sieve or paper towel/kitchen towel — to check for any particles that have not broken down.