Those are indeed classics that I remember well lol. Out of curiosity, with western game developers really booming over the past decade+, has there been any such cases that happened with shoddy localization into the Japanese market? Do you find that western games are localized well, or did they go through a period of infancy as well?
This is getting into the realm of personal opinion, but I think that the video game industry as a whole does a much better job of translating games INTO English than we do of translating games FROM English. Translations From EN to European languages tend to be higher quality than those to distant languages like Japanese and Korean, but there are many, many examples where mistranslations occur even with languages that are "closer" to English. I don't feel it's appropriate to call out any specific examples, but if you're interested in the topic, a quick google search will provide you with a plethora of mistakes and missed opportunities.

Speaking from a more general perspective, English is a relatively risk-free language to translate into. You can just go with American spellings and language conventions and people will understand it quite easily, even if they come from non-American English speaking backgrounds, due to the permeation of American culture into the global mindset. Spanish, French, and Portuguese are very risky languages, because the discrepancies between the European and North/South American dialects are high enough that players feel a huge of disconnect if they play a version intended for another region.

Sometimes I will play a Western game with a Japanese language pack just for fun, and the translations are really all over the map. Some are very professionally done, with excellent nuanced localizations; some have outright mistranslations that impede the player's ability to progress; and others are borderline incomprehensible. We've still got a ways to go.

Your bit about actually targeting a western audience with the localization, rather than just a translation, is really interesting. I've never really thought much on it before, but it makes a lot of sense the way you laid it out. Looking back, I've certainly read/played my fair share of media that hadn't had such care given to it. Looking at some recent quests as examples, Saved by the Bell, To Catch a Predator, I'm on a Boat, Don't Ever Leaf Me, Hide and Go Peak, and many others, make sense through English and our culture, they're phrases that we know from life, shows, music, etc. Out of curiosity, do quests like these take on relevant names that may invoke similar "I know what that's from!" feelings for the Japanese community? Any examples to share that we might know?
A lot of this comes down to player expectations and translator/writer preference. Even in the Japanese, some of our planners prefer light, playful names, while others like something more serious, so it's not a set rule. The Tarutaru Sauce questline for example: The planner always uses a play on words by replacing one of the kanji in the quest line with the kanji for "to laugh." The planner for the Svenja questline plays it straight. Because the Japanese is relatively freeform, we make things light hearted in the English whenever we feel we can fit in something good that fits the story.

Is there any chance that someday you may be able to give a bit of a description to some auto translate terms that are ambiguous, or that are misused often? I'd be interesting to know what they actually say/mean in Japanese. "Please use a weak weapon to attack" seems straight forward enough, but I've had a few Japanese players use that in party (not during something like Void Watch) and I was left confused at what they were getting at. Some terms of our's that may not be well known is "Excuse me..." "Are you alone?" "Can you hear me?" and many others. They seem straight forward enough, but whether due to the language barrier being too much of a bother to communicate through, or our knowledge of what the terms come out to be on the Japanese side, it's difficult to get a clear answer from Japanese players.
Ohhhhh I can give you a very good one. I was talking with the planner in charge of the auto-translate dictionary the other day about this very topic. He remembers one instance in particular where he was just chatting with his linkshell, sitting in Jeuno and someone in /shout was looking to buy some item or other:

"{Please} {Reward} 300k"

The problem is that the player selected "Reward" from the Job Abilities category, not the Trade category, so it popped up in Japanese with the beastmaster ability, which in Japanese is "Itawaru"--a verb rather than a noun. "Itawaru" means "to treat someone nicely," so the guy was basically sitting in /shout going "Please treat me nicely 300k."

Long story short: make sure you took before you translate!