I'd also suggest that any multi-angled sprite games (that weren't pre-rendered) were essentially pixel art. I think for Doom they actually used photos of physical models for some sprites, but even then there must've been a lot of direct pixel work happening.
I agree (And as far as the process goes, this was exactly the case in
Doom), and I think that considering them non-pixelart, is something that comes from the modern-idea of pixelart - today's limits are self-imposed, yesterday's limit's were actual hardware limits - if I could suddenly work with 256 colors instead of 16 for example, I'd do so

I'm sure the artist did use their tools to their full potential, and pixelart of the days used both the approach we use today and index-painting as it's called.

Quick compat between EGA and VGA (Doom Palette) - personally, I'd like to use most of the colors to make the transitions as fluid as possible and it's just tedious to do it by hand on every pixel (unless it's a specific detail part). Actually, since most of the 2,5d games had the dark-fog effect for atmosphere, and some transparency (using transparency tables) as standard, limiting the color palette would make them ugly, which obviously wasn't desired

Here's a sprite from Wolfenstein, drawn from scratch if you ask me.

Either way, you still have the whole character to draw.
I think most games used prerendered sprites or models, eventually with some hand touchups, because it's a bit less of a workload - pixelling a character walking, attack, pain and death frames ( I guess it's at least ~10 unique frames ) in 5/8 directions (depends if the character is symetrical) is quite a bit

Some of the stuff can be copied and used (for example head and torso) but the rest has to be drawn in from scratch. I've animated a character like this digitally painting and it took a while

Bigger color palette also made it more attractive to use prerendered 3d artworks which was quite a hype back then

while pixelart wasn't anything special - it was a standard of working with 2d GFX back then.
Creating a 3d game with pixelart graphics, is a tremendous workload. Some examples of such projects, and what limits they've used to lighten the amount of work a bit:
Amiga:
http://hol.abime.net/512/screenshotFears Amiga first person shooter - it uses pixelart for it's gfx, and of the cool amiga-style variety

The characters are one sided though!
http://hol.abime.net/2739/screenshotGloom's enemies have more directional sprites I think, but notice they are pretty simple though.
As far as modern stuff goes, there's exactly the same two approaches:
1. Keep it simple stupid

Aforementioned Minecraft, which is really simple in terms of GFX (I'd say it's even primitive, since the pixel tech used for it's GFX is pretty low-level)
Or the idea of totally oldschool, 8-bit console look - for example the MegaMan 8Bit Deathmatch:

Lower res sprites, color limits - you can finish your sprites a lot quicker
2. Establish some kinds of limits - either directions or the amount of frames in animation:
Retroblazer - uses single direction for it's enemies, judging from what I've seen (correct me if I'm wrong!)
(Oh and not everything is pixelart there, IIRC)