As a very general rule, bass is the most subjective part of the EQ and has no particular advantages or disadvantages. Some people like their low strings to be loud, others don't. Usually if you're tuning low or using very thick strings, which will naturally produce more volume, you'll want a pickup that has less bass to balance the sound. 6-string electric guitar tuned to E Standard doesn't produce much in the way of bass frequencies so this is the part of the EQ you'll notice the least if you're not tuning low.
Mids are often the most important are of EQ and, personally, I like to break them down to 'low-mids' and 'high-mids', since just 'mids' covers such a wide range of frequencies. The lower-mids are where you get the 'punch' when you hit a powerchord and is what stops single notes around the middle of the fretboard on the 4th and 3rd strings from sounding weak. Low-mids are the friend of rhythm players but can make a pickup seem 'muddy' to fast-picking, high-gain lead types. The high-mids are the reverse of this, giving you most of the guitar's clarity and making any single notes really jump out. Where low-mids 'punch', high-mids 'cut'. Whenever you hear someone talking about 'cutting through the mix' or similar, they're talking about the upper half of the mid range. Have you ever seen an amp with 'resonance' and 'presence' controls? Those typically (not always) work by adding back in the lower-mids or high-mids, respectively, after the preamp section of the amplifier. You can think of a pickup's mid response in the same way: the lower half of the mids gives a pickup more resonance while the upper half gives it its presence.
When a manufacturer or random person on a forum only talks about 'mids' being strong/pushed or weak/scooped, without specifying whether they mean the higher or lower mids, it doesn't really tell you much since scooping the higher mids is a very different sound to scooping the lower mids, and the same is true when one part of the mids is pushed. If all the mids are being pushed/scooped, which tends to only be the case with really high-output pickups with very closely-matched coils, you get either a hollow, thin sound (scooping every mid more than treble or bass) or you get a kind of 'AM radio' tone (pushing every mid more than treble or bass). Cooler pickups and especially ones with mismatched coils have a very hard time affecting all of the mids equally and instead tend to favour either the lower half or upper half.
Whether lots or reduced mids (of any type) is better varies wildly depending on the amp, amp settings, speakers, and of course playing style. With some rigs having a lot of mids gives you clarity while with others you only get clarity by having less mids. The balance of low-mids/high-mids/all mids, and the rest of the guitar and amp set up is too complicated to possibly cover every scenario here. As another very general rule, getting a pickup that has its mids balanced opposite to your amp and the guitar's construction is how you get more clarity, whereas if you want to scoop your mids at your amp you're usually best off having pickups that produce a lot of mid power. But, as I said, it really depends on which part of the mids we're talking about and... y'know, I'm just going to end this part there, or I'll be going on for days. The point is, the mids are the most important part of the pickup's EQ since that's where most of the guitar's frequencies are found, but whether that means you want a lot or a little comes down to every other aspect of your music.
The treble you get from a pickup is often one of the more obvious characteristics, since guitars are capable of putting out quite high frequencies and a lot of modern amps reproduce high frequencies more accurately (i.e. loudly and clearly) than older amplifiers, especially if you compare (most) transistor amps to (most) valve amps. (The same is true of recorded music. One of the downsides of vinyl is it simply can not reproduce the extreme treble and bass that digital can cover, but the upside is a lot of people find that lack of 'extreme' treble to be more comfortable to listen to.) Personally I believe this is why so many people these days are only asking to either get more treble or smooth off the treble and few ask about bass.
If you play a lot very high up the neck on the 2nd and 1st strings then you're going to want a pickup that people rate for strong treble, since those screaming high notes can't really be heard if your pickup doesn't react to them well. Strong treble is also needed if you want each note in a full chord to ring out clearly. Conversely, some people want the notes in their chords to smear together into a single solid sound, which is where treble-deficient pickups like the Invader come in. Like bass, how strong you want your treble is highly subjective and isn't as reliant on your amp (or pedals, etc) as the mids are.
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