Astronomy, in a few hundred words ...
The sky has now been 'imaged' from ~TeV gammas to ~1 MHz radio waves.
The angular resolution is ~1" (or better) in the (soft) x-ray band, UV through to mid-IR, and also in parts of the radio spectrum. For most of the rest it's ~1' (or worse). Objects smaller, in angular size, than the resolution are 'point sources'.
A point source in one wave-band (the optical, say) can often be confidently matched to a point source in another (mm microwave, say), by astrometry - the estimated location of the 'point' on the sky (in RA and Dec) can usually be estimated with an uncertainty ~10 times smaller than the resolution. Of course, this doesn't always work ... within the 'point' in some gamma ray band there may be a dozen or a hundred x-ray or IR point sources, or even two or more extended sources.
Often a point source in one wave band has no matching point source in any other wave band; if the observations have been done carefully, this enables you to make estimates of the upper bound of the 'brightness' of that point source in the other wave band(s); later observations, made with more sensitive instruments, might detect something of course.
Leaving aside solar system objects (easy to categorise as such by their movement over time periods of seconds to hours) and transients (point sources, in any wave band, that were not there yesterday, here today, and not there again tomorrow), what are the billions of point sources that astronomers have catalogued?
A great many are 'stars'; many are 'quasars'; some are 'galaxies', 'BL Lac objects', 'supernova remnants', 'globular clusters', 'HII regions', and so on.
What is the basis for classification? Here's where it starts to get interesting!
When a new class of object in the sky is discovered it is classified strictly by what is directly observed - its brightness, its colours, its spectrum, it variability, its environment, and so on. GRB (Gamma Ray Bursts) are a good, recent example.
The race is then on, among astronomers, to understand what this new class of object 'is', 'in reality'. Before too long a consensus emerges, and the classification shifts to the underlying model; the many classes of variable stars show this very nicely - the class name (e.g. 'RCrB', or 'R Coronae Borealis' stars) reflects the name of an object that is regarded as the standard (often the first, historically, to be discovered) and the class is defined, observationally, by its light curves, colours, etc, but the term now refers ultimately to 'the reality' ('These are rare, luminous, hydrogen-poor, carbon-rich, variables that spend most of their time at maximum light, occasionally fading as much as nine magnitudes at irregular intervals. They then slowly recover to their maximum brightness after a few months to a year. Members of this group have F to K and R spectral types', in this case).
So too with 'quasars' ... or 'quasi-stellar radio source': the first objects to be observed were, indeed, 'quasi-stellar' (meaning they are point sources, in the optical wave band, as observed by telescopes of the time ... the HST hadn't been invented then), and 'radio sources' (meaning they had been detected as point sources by radio telescopes of the time). Soon, however, it was discovered that some 'quasars' were not quasi-stellar (there was some 'fuzz' around them, even in the Palomar Schmidt plates; at higher resolution in the radio, many turned out to be 'double-lobed'), nor 'radio sources' (they had spectra and colours and time variability that were similar to 'true' quasars but no radio source was detected, at the time). Later it was discovered that many quasars resembled the nuclei of certain galaxies (Seyferts).
And so on, through Lyman forests, x-ray sources, IR sources, jets, BL Lacs, and much, much more.
To distinguish the wide range of different kinds of objects, various sub-classes were defined, like QSOs (quasars without the radio), OVV quasars, type 2 quasars, RLQs, RQQs, ... Among astronomers, in the papers they wrote, this plethora of definitions and (sub) classes occasionally caused confusion, but as long as papers were written with the expected care, such confusion was quite limited.
Not so for the general public, and doubly not so for those not trained as astronomers (or almost any branch of science; the issues of classification and definition are pretty similar) ... and especially for BeAChooser (and some others)!
So, what about today? What's a 'quasar' today?
First, make sure you understand how the author(s) of a paper is/are using the term; that usage is primary.
Second, in general, a quasar is an AGN (active galactic nucleus; this Wiki page is a good start, provided you take on board all the usual caveats about Wiki pages), or is the accretion disk plus super-massive black hole at the centre of an AGN.
Questions?