It started with the NES. Yes, with the NES. The NES and SNES had great 3rd party support but the hardcore PC and SEGA fans knew what was really up.

Check out this Nintendo is Doomed newsgroup post from 1990:

"Are they going under like Mattel, Coleco, and Atari?? I see their
games for $9-$25. Very few are over $35 anymore. I can't believe that
new games like Skate or die 2 are only $18 at Electronic Boutique.
Also, everyone says they saw that Nintendo was coming out with the
16 bit Super Famcom in January. A video dealer here says it was in a trade
magazine. If this is even close to being true, why does Nintendo deny
any plans of a U.S.A. version? It seems that they would ad to the rumor so
prospective 16 bitters will not buy a Genesis but instead wait for the
SF. I say the hell with Nintendo, Genesis here I come"

https://groups.google.com/forum/?fromgroups#!topic/rec.games.video/IKQkExqofSA

Maybe its semantics, but popular with thirds doesn't mean they were liked by them. The NES was hated by thirds most notably EA who's board had to threaten Trip Hawkins to put games on the system due to its popularity. Also because of Nintendo's strict stance on cartridges and quality control, thirds weren't able to flood the market with tons of crappy software. Once Sega came onto the scene, EA jumped behind them almost exclusively to try and oust Nintendo from the console market. But What many hadn't thought at the time was that Sega took a similar stance to Nintendo on quality control and wasn't able to leverage Sega as they'd wanted.

When Sony came into gaming with their deep pockets no quality control, along with not being a 'gaming company' Thirds were able to have their investments leveraged as they were now able to control a console maker.  

Really since the PS1 era. A lot of people were just happy to see the "giant" fall. PS1 was the system, Dreamcast was the unappreciated genius, and the N64 was mostly considered awful outside a few first party must haves.

Which is basically every Nintendo System since then as far as your average super involved internet user goes.

It's also when the Nintendo consoles stopped getting multiplatform games.  the Cartidges were a big reason and it just continued from there.

I don't think Nintendo has always been the red headed step child. I think Nintendo has always done their own thing. When they had the majority of the market, it was great. When they were censoring games like Earthworm Jim or Mortal Kombat, it wasn't as big a deal. Nowadays, it kinda stands out more because the market is so different. Nintendo always did what they wanted to do and that's it. Now, you have companies like Microsoft and Sony that do what they think the people want.

Probably after SNES. Basically ever since the Playstation came along, which ended up becoming a fan and developer favourite.