I agree with JKos, 'balanced' is used in several different ways by people and string manufacturers, at least the 4 ways JKos listed, and the various combinations of those, so one has to first specify how the word is being used.
I want to avoid having one string as taut as a hunting bow and have another be a wet noodle in comparison (exaggeration, I know).
So in this case you seem to be referring to 'scientific tension', the force value in 'lbs' (pounds) found on tension charts.
But you might instead be referring to 'perceived tension'/'feel', the 2 are different.
Are you concerned with the actual scientific force each string exerts on the headstock, or 'how tight each string feels to the hands'?
Well, "balance" obviously may refer to two different things: sound and feel.
You missed out probably the most common meaning: 'scientific tension', which is different to 'feel'.
More recently, since Circle K Strings helped to popularise the concept of 'equal tension sets', 'balanced' is increasingly used to mean 'equal scientific tensions'. This is the meaning D'Addario uses for its 'balanced' sets.
(measurable tension is not elasticity. Feel is determined by the latter. Not to speak of gauges which won't tell you either.).
Gauge is a fairly reliable but approximate indicator of scientific tension, your statement implies no relationship.
Try playing open strings with hammer-ons on the 3rd, 5th and 7th fret, both acoustically and amplified. Their volume, sound and feel should be nearly identical. Also switching between various techniques (pizz, pick, slap, doublethumbing, touchstye) should sound plausibly good (four your individual taste). Then play across the fingerboard, both vertically and horizontally.
That is only one very specific combination of aspects as a definition of 'balanced'.
I always wondered how people claim a given set is not balanced. It's as if they can just tell from looking at them. I suspect some people are BS-ing.
If one assumes the meaning is 'scientific tension', then you can look at the gauges which are an approximate indicator of scientific tension. For example .107 .080 .060 .045 is probably close to having equal scientific tensions, there is a way to approximately tell by looking at the gauges (there is a rough rule that can be derived scientifically, as i have done).
But, gauges are not an exact indicator of scientific tension, so this is an approximate guess until one can look at the actual measured scientific tension values on a tension chart.
But it is valid to look at something like .105 .085 .070 .050 and say 'that is unbalanced' (in terms of scientific tension) because that indicates a very high probabllity of a very top-heavy set. So it is not necessarily 'BS' =)