Welcome to roleplaying! I know it can be daunting; there are literally thousands of RPGs on the market as well as out of print ones that people still play.
What is roleplaying?
Many a roleplaying game has a "What is roleplaying?" section in the front, and they all have different takes on it, but the most common summary is that it's a formalized version of the kids' game of "cops and robbers." Though trying to generalize how role-playing games work is kinda like trying to generalize how board games work (there are many exceptions to every statement you could make), most commonly you create a "character" that represents someone in the fictional world - a wizard or cowboy or space trucker or whatnot. Their attributes - what they look like, how strong and fast and smart they are - is written down on a "character sheet." You are part of a group of other players who also have characters, this is usually referred to as a "party." There is a Game Master who doesn't usually play a character of their own, instead they depict the whole world and all the people in it who aren't player-controlled characters (NPCs - non-player characters).
The game has rules, and usually dice, which are used to determine if you succeed or fail at difficult tasks. If you are Legolas trying to shoot an orc with your bow, you would roll a die and consult your character's skill and the result would determine if you hit him or not, for example.
As a player, you run a character (like in World of Warcraft or the like) and go about on adventures of whatever type is appropriate to the game's genre. So in this way, using both game rules and the participant's imagination, you can all participate in an exciting story.
Some games don't use dice, don't have a game master, don't have you play one specific character you identify with, etc., but mostly they do.
Game Genres
What kind of story? Well, that's one reason why there's 1000 RPGs on the market. There's everything from stock fantasy (Dungeons & Dragons) to stock science fiction (Traveller) to games that are specifically licensed properties from TV shows or movies (Serenity, Dresden Files, Ghostbusters) to really unusual concepts, like Mormon cowboys keeping the peace (Dogs in the Vineyard). For any genre - superheroes, pulp, horror - there are many games on the market.
Game Complexity
Some RPGs are quite complex and require loads of rules and multiple books to play. Some people like this approach, and it can be more like miniatures gaming or complex wargaming. Some games are very small and have extremely light rules and focus more on the story and imagination aspect, and verge towards being collaborative storytelling games.
Recommendations
The most played game is Dungeons & Dragons, in its various incarnations. Because you need a group to play (though you can play online, and there are games tuned towards just two people - some are even for one person, though these are more just choose-your-own-adventure books) it can help to choose a popular game. D&D, especially in its modern incarnations, definitely is one of those multi-book complicated rules heavy games, though, so it can be an intimidating starting point. They usually have a starter set with simplified rules that you might consider using to kick the tires on the concept.
Though if you are willing to start a group of your own, maybe just by corralling a couple friends to play, you really can choose whatever game you want. Then, a less rules-intensive game might be a good choice. Pick a genre you like and there's probably a good starter game in there - ICONS for super-heroes, for example. Or Call of Cthulhu for horror. A game that has a good amount of published adventures a GM can run for their players is a good place to start because creating your own adventures can be hard when you're just getting started, and it'll help you get the feel for how others do it. There are gaming stores in most major cities (often comboed with comics or board game stores) and you can go browse at your leisure.
Find Other Roleplayers
In the end, the best way to get started is to find someone who roleplays already - you probably have a friend that does and you don't know it. See Where can I find other RPG players? on how to find other gamers. You can try it out with them and learn more and see if you like it.
Next Steps
As a more in-depth primer to pick up where this leaves off, there is an excellent pair of short PDFs called How to Play Roleplaying Games and How to Run Roleplaying Games. They provide a very nice "what does it actually look like and how do I make it happen?" primer for new players and new game masters.
It's a fun hobby, many of us have been doing it for 20+ years. Check it out, it can be a blast!
This entirely depends on your game, and your gaming group.
What you are referring to (acting as yourself instead of your character) is called "meta-gaming" and involves making decisions that are outside of the purview of your character's personality or knowledge. Whether or not this is an acceptable practice depends in part on your group and in part on the level or realism/seriousness of your game.
If your group is a lighthearted, we are playing this to have fun type group and you are playing in a system that focuses more on mechanics and tactics than on actual role-playing, then meta-gaming may be ok. You are likely free to make, at least in some limited sense, decisions that would not be likely/possible for your character to make, either because you personally feel a certain way, or because you have knowledge your character does not.
However, if you are playing in a more serious group, or more realistic setting where role-playing is the primary influence then it would be inadvisable to make decisions your character would not make, or act on knowledge your character does not have.
These are not discrete decisions either. Most groups fall somewhere in the spectrum between heavy story influence and heavy tactical influence. Systems do as well, although I feel this is more highly dependent on group dynamics and expectations than it is on the system you are playing in.
Sometimes role-playing your character leads to suboptimal (even abysmal) outcomes, this goes somewhat against our typical gamer mentality (I know it does mine), but is part of what makes RPGs special. You can have a ton of fun, even when your character dies horrible.
Talk to your group about what their expectations are. This can dictate a lot of how you role-play and how you make decisions. Does your group expect you to play your character within a certain set of boundaries (emotionally or mentally), or is this a no-limits, no holds-barred kind of group? This is something you can bring up with your group at the beginning of a session: "how much do you guys expect internal consistency on an emotional/knowledge level do you expect?" (or "is meta-gaming appropriate at this table?")
Remember that the goal of playing an RPG, most of the time, is not to win, but to tell a good story, whether you're role-playing your character or meta-gaming, remember this purpose.
Best Answer
Check out the PHB
The PHB approaches the game more from the player side while the DMG approaches it from the GM's side. This means, generally speaking, roleplaying descriptions in the DMG are focused on NPCs and how GMs can roleplay them effectively, while roleplaying descriptions in the PHB are more focused on how players can effectively play their characters. The second quote from the DMG provided in the question is from a section to help GMs recognize what kinds of players are playing. If you read the entire section on page 6 of the DMG it is more describing a player type than a method of roleplaying or acting.
There are some suggestions in the DMG for rewarding players for inspired role playing, for example the Inspiration rules starting on DMG 240. I find Inspiration points to be fun and helpful with both beginners and advanced players.
Descriptive and Active Roleplaying
For a good description of Descriptive Roleplaying and Active Roleplaying, which may be what you are looking for, check out PHB 185. Under the heading Roleplaying the PHB states:
Descriptive Roleplaying
Later on PHB 186 under the Descriptive Approach to Roleplaying header the PHB offers this summary (after a more detailed description):
This sounds similar to the first quote in the question and may be what you were thinking of.
Active Roleplaying
Active roleplaying is more like acting and shows the other players and GM what your character is doing rather than telling them what your character is doing. PHB 186: