There are definitely levels of weeaboos
The type the second picture describes is like the worst it can get and probably what most people think of when they think of weeaboos. Any who watches anime in general gets called a weeaboo, even though that's probably not necessarily the case. I do like to call myself a weeaboo though, just as an easy way to label myself generically, but that's probably not the best idea. My idea of weeaboo is just someone who likes anime and stuff, but deeper weeaboos - as I mentioned earlier - are like the second picture.
I do appreciate Japanese culture, though. It's like their culture has been so preserved throughout time that it just comes off as the most "out there" culture to us. If I had the opportunity, I would've gone to like a kyuudou club since. It seems very graceful and meditative and bows are cool to me. I like Japanese food, I would've liked to take kendo as well, since it seems like a really interesting sport. Much more interesting than anything they have here in the States.
Does that make me a weeaboo? Probably. I do prefer aspects of Japanese culture over my own (American, I guess), but I wouldn't say I'm obsessed with it. I'm not going to dress up like an anime girl and start calling all my seniors senpai.
If we define weeaboo as the way everyone thinks it is, a fat neckbeard who dresses up in a kimono and has bodypillows and a samurai sword (basically your second picture), then there's something wrong. There are things wrong that I probably couldn't even begin to understand since it probably has a lot to do with their mental state. How bad does it have to get to throw away everything around you to imitate a culture you'll never truly be part of? You're probably cringy, even to them. It's like becoming the embodiment of an exaggeration of a specific aspect of their culture, and you just stick out like a sore thumb.
But if we talk about a weeaboo as someone who just appreciates Japanese culture and isn't a "deep" weeaboo, then I don't think there's anything wrong.