The Update/Social/Promotion/Forum thing are part the Priority Inbox feature, which you can turn off if you don’t find it helpful.
As for Inbox, yeah, it’s just another label – one meant to let you help you organize whichever emails are still current/relevant. Once you’re done with something you should Archive it (or X out the inbox label, same thing) and then it gets moved to the All Mail label.
The labels mostly function like folders except that each message can have more than one label. And when you say “search is inefficient”, I mean, they all take less than a second to show up anyway so it hardly matters. In terms of behind-the-scenes computer stuff, they’re just entries in a giant database distributed across thousands of supercomputer clusters. You can call them whatever you like, but in the end you ask for X and Google’s servers find it for you in half a second. It’s not like they are actually putting your emails into MS-DOS folder structures; it’s ALL an amazingly indexed, supremely optimized, and massively parallelized operation in some database of theirs.
I dunno if an analogy would help make it clearer, but imagine a bunch of envelopes coming in to your mailbox.
Your intern (or dog) brings them in and dumps them on your desk (this is your Inbox).
Your secretary uses four different colored post-it notes to let you know which envelopes are from marketers, which are from your personal friends, which are from other companies, etc. These are the Priority Inbox labels (Social/Update/Promotion/Forum).
Then you go through them yourself and add your own post-it notes: This letter’s from Mom, that one is from your favorite charity, that one is an angry rant from the IRS. These are your own personal labels.
If you’re the lazy kind of guy, maybe you just read the letters and leave them sitting on your desk. That’s akin to leaving all your messages in your Inbox.
If you’re a bit more organized, once you’re done with a letter you move it into a big filing cabinet where all your mail sits. All the post-its are still there, but it’s no longer on your desk (Inbox) – Archive just stores the old mail without having it clutter up your inbox.
If you’re super-organized or just paranoid, maybe you don’t want to store old mail. Instead, you just dump them in the recycling bin. That’s what the Delete action does. Whereas Archive keeps the mail but moves it out of the inbox (so it’s still searchable in the future), Delete gets rid of it altogether.
Now, why don’t you just use filing folders instead of post-its, you ask? Well, what if you’re obsessively organized and you mark up letters as follows: “Personal correspondence” “Friends”. But then later on Mom gets involved in the same email thread, so you mark it as “Family” as well. And your sister adds cat photos, so you label it “Funny” too. Five years later, you’re looking for that email… where was it? If you used folders, you’d have to go through Family, Friends, and Funny until you found the one right folder. If you used labels, any of them would bring it up. (That’s the theory, anyway. In practice I just use them like folders, and most people I know do so as well.)
What on earth is the Update Label for???