It's definitely a different game when white doesn't close the center and go for the "classical" pawn storming lines. While those lines are often the easiest to play (black can play f5, f4, g5, g4, g3 without thinking too much), those are also the lines that are arguably the most challenging for black to face.
When white keeps the tension in the center, black has to decide when to break that tension. Generally speaking, black should do everything possible to develop before playing exd4, as this move nearly always cedes white the entire center. If black is going to turn over the center to white, it's vitally important to be able to immediately counter attack. This means that moves like c6, Qc7, and Nbd7 should be played before committing with exd4. The plan with Nc5 pressuring e4 is often best, but make sure that you can support the Nc5 with a move like a5, preventing b4 chasing the knight away. The queen on c7 is often useful because it allows the rooks to come to e8 and d8 where they defend the backwards d6 pawn and pressure e4. Black's plan in these structures is almost always to push d5. The light square bishop can support this push from e6, but if the white knight stays on f3 then ...Be6 can be well met by Ng5.
In conclusion, these structures are very double edged and black must be precise in order to maintain equality.
When white takes on e5, black usually can achieve a very good game with some precision. The trick here is to notice that black can use the d4 square, but white can rarely use the d5 square because black can always post the c pawn on c6 and keep pieces out forever. A common plan is to rearrange the black pieces with Qc7, Rd8, Bf8, and some lengthy maneuver with the f6 knight. This is often ...Nf6-d7-f8-e6-d4. The black bishop is frequently left on g7 until the knight has reached d4, and only then is it transferred to the queenside with ...Bg7-f8-c5.
These plans are very common, but they're also topical. If white is trying to prevent one or more of these re-positioning efforts, then black will have to use tactics in order to justify the moves.
It's very hard to explain quickly, but these plans are usually much easier to play in a long game when you can dedicate dozens of minutes to planning the maneuvers and then carefully execute them. It's well worth the time investment to play long games with these structures, even if you don't want to make the KID your primary black weapon against 1.d4.
