What I mean by 'invention': Well, I'm not the most gifted wordsmith, to be sure. When I speak of such concepts as love, good, evil, mistake, logic, reason, etc., I'm speaking of linguistic terms humans have invented to cover things that are not themselves clearly understood by common folk. Love, for example, is an invention; it is a term used to describe a very complex set of biochemical reactions and the brain's subsequent processing thereof, in relation to an individual's interactions with another individual (or, of course, with itself). 'Love', then, becomes a term of convenience. If you want to say love is an 'illusion', I could certainly agree.
The same goes with most of these non-concrete ideas we're discussing - and a 'mistake' is one of those ideas. Physical processes happen; it takes a sufficiently complex computational system to assign one of those processes or set of processes the label of 'mistake'.
A synonym for 'mistake' is 'error'. Computers certainly are capable of making quite a few errors. Strangely, the more complex we make our machines, the more prone to error they become. No machine is more complex than the human brain, and the greater complexity also results in a greater propensity for error. Our brains have developed to allow for much greater flexibility and wider array of possible actions, resulting in brains that are 'jack of all trades, master of none'. They are vast organic computers with complex multi-parallel processors that take shape as we learn, creating networks of connections that often result in strangely related information processing (as the joke goes, 'what do windshield wipers have to do with breasts?').
Anyway, the point you kept harping on, 'can machines make mistakes'? The answer is, absolutely - in the same way machines can be beautiful or fearsome or useless. In other words, a mistake is a human label to apply to any set of unexpected and/or undesired behaviors of a system. If I have a color-by-number book and I rely on my admittedly poor memory to color it, I might 'make a mistake' and color the 3s red, when they should be blue. Mechanistically, there was no particular 'mistake' involved, but due to the inventions of 'instruction', 'language', and 'mistake', I made a mistake. On the other hand, if my intention was to color it red - for example, I felt that a red race car was more interesting to me - then I made 'no mistake'; yet others might still feel I made a mistake. It is, as you suggest, all illusion. It is also human invention.
Humans invent all sorts of convenient illusions, such as politics, religion, morality, superiority, subservience, beauty, and so on and so forth. All illusions; all inventions.
A light bulb is an invention, yes. It is also an illusion. It is a mass of atoms and molecules that, when electrically charged to the proper degree and along the proper pathways, stimulates emissions of photons. We have invented terms like 'light' and 'bulb' to simplify communication about this object; these terms are illusions as well, if we want to look at it that way.
So, to sum up: a mistake and a light bulb are both inventions and illusions. Logic, likewise, is an invention and an illusion, as you use the term. Machines are capable of doing logic, unless you can prove either a) that no machine anywhere ever can do logic (proving a negative - not possible), or b) that those machines that are known to do logic are not simply machines (prove that humans possess immaterial (= in no way related to matter or energy) qualities - due to definition of 'immaterial', not possible either).
I think this discussion really can't be completed satisfactorily, until such time as the brain and its processes are fully understood; such time as a manufactured entity demonstrates full philosophical reasoning ability; or until such time as a means is determined for detecting and analysing things which violate the principles of materialism. Until such time, we're deeply in the realm of conjecture.
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