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My question was:

The JSFiddle demo is here, but am pasting the code below also. My question is simple (though I can't seem to find any example in how to realize this): right now if you start the game in JSFiddle you will see that the rectangle immediately "falls" down. I would somehow like the game to start when I click with the mouse (basically that nothing happens before the first click of a mouse) - any points on how to accomplish this?

JS:

// Initialize Phaser, and creates a 400x490px game
var game = new Phaser.Game(400, 490, Phaser.AUTO, 'game_div');
var game_state = {};


// Creates a new 'main' state that wil contain the game
game_state.main = function () {};
game_state.main.prototype = {

    preload: function () {
        // Change the background color of the game
        this.game.stage.backgroundColor = '#71c5cf';

        // Load the bird sprite
        this.game.load.image('bird', 'https://lh6.googleusercontent.com/Xrz0PnR6Ezg5_k5zyFKxGv0LzehAP9SMj_ga3qQzIF4JAfv8xHm7TxfliwtBD8ihfw=s190');
        this.game.load.image('pipe', 'https://lh4.googleusercontent.com/RSMNhJ3KY4Xl0PQpUf6I9EayOdLhvOKKV9QV7_BXXYVedPy0oMNRFKANW14xV76NDA=s190');
    },

    create: function () {
        // Display the bird on the screen
        this.bird = this.game.add.sprite(100, 245, 'bird');

        // Add gravity to the bird to make it fall
        this.bird.body.gravity.y = 1000;

        // Call the 'jump' function when the spacekey is hit
        var space_key = this.game.input.keyboard.addKey(Phaser.Keyboard.SPACEBAR);
        space_key.onDown.add(this.jump, this);

        this.pipes = game.add.group();
        this.pipes.createMultiple(20, 'pipe');
        this.timer = this.game.time.events.loop(1500, this.add_row_of_pipes, this);

        this.score = 0;
        var style = {
            font: "30px Arial",
            fill: "#ffffff"
        };
        this.label_score = this.game.add.text(20, 20, "0", style);

    },

    update: function () {
        // If the bird is out of the world (too high or too low), call the 'restart_game' function
        this.game.physics.overlap(this.bird, this.pipes, this.restart_game, null, this);
        if (this.bird.inWorld == false) this.restart_game();
    },


    // Make the bird jump 
    jump: function () {
        // Add a vertical velocity to the bird
        this.bird.body.velocity.y = -350;
    },

    // Restart the game
    restart_game: function () {
        // Start the 'main' state, which restarts the game
        this.game.time.events.remove(this.timer);
        this.game.state.start('main');
    },

    add_one_pipe: function (x, y) {
        // Get the first dead pipe of our group
        var pipe = this.pipes.getFirstDead();

        // Set the new position of the pipe
        pipe.reset(x, y);

        // Add velocity to the pipe to make it move left
        pipe.body.velocity.x = -200;

        // Kill the pipe when it's no longer visible 
        pipe.outOfBoundsKill = true;
    },
    add_row_of_pipes: function () {
        this.score += 1;
        this.label_score.content = this.score;
        var hole = Math.floor(Math.random() * 5) + 1;

        for (var i = 0; i < 8; i++)
        if (i != hole && i != hole + 1) this.add_one_pipe(400, i * 60 + 10);
    },
};



// Add and start the 'main' state to start the game
game.state.add('main', game_state.main);
game.state.start('main');

CSS:

#game_div {
    width: 400px;
    margin: auto;
    margin-top: 50px;
}

HTML:

<div id="game_div"></div>

I found the answer to this one myself:

So, to not leave this question unanswered and to help further readers, here is the link to the updated jsFiddle where I solved my initial problem, and below is the code from jsFiddle.

However, if one will be interested I suggest taking a look at the freely available code on GitHub which uses game states and has a better structured code.

jsFiddle

CSS:

#game_div {
    width: 400px;
    margin: auto;
    margin-top: 50px;
}

HTML:

<div id="game_div"></div>

JS:

// Initialize Phaser, and creates a 400x490px game
var game = new Phaser.Game(400, 490, Phaser.AUTO, 'game_div');
var game_state = {};
var gameStarted = false;


// Creates a new 'main' state that wil contain the game
game_state.main = function () {};
game_state.main.prototype = {

    game_start: function(){
        if (!gameStarted){
            this.bird.body.gravity.y = 1000;
            this.timer = this.game.time.events.loop(1500, this.add_row_of_pipes, this);
            this.label_start.content = "";
        }
        gameStarted = true;
    },
    preload: function () {
        // Change the background color of the game
        this.game.stage.backgroundColor = '#71c5cf';

        // Load the bird sprite
        this.game.load.image('bird', 'https://lh6.googleusercontent.com/Xrz0PnR6Ezg5_k5zyFKxGv0LzehAP9SMj_ga3qQzIF4JAfv8xHm7TxfliwtBD8ihfw=s190');
        this.game.load.image('pipe', 'https://lh4.googleusercontent.com/RSMNhJ3KY4Xl0PQpUf6I9EayOdLhvOKKV9QV7_BXXYVedPy0oMNRFKANW14xV76NDA=s190');
    },

    create: function () {
        // Display the bird on the screen
        this.bird = this.game.add.sprite(100, 245, 'bird');

        // Add gravity to the bird to make it fall


        // Call the 'jump' function when the spacekey is hit
        this.game.input.onDown.add(this.game_start, this);

        this.pipes = game.add.group();
        this.pipes.createMultiple(20, 'pipe');


        this.score = 0;
        var style = {
            font: "30px Arial",
            fill: "#ffffff"
        };
        this.label_score = this.game.add.text(20, 20, "0", style);
        this.label_start = this.game.add.text(35, 180, "Click to start the show", style);
        game.input.onDown.add(this.jump, this);
    },

    update: function () {
        // If the bird is out of the world (too high or too low), call the 'restart_game' function
        this.game.physics.overlap(this.bird, this.pipes, this.restart_game, null, this);
        if (this.bird.inWorld == false) this.restart_game();
    },


    // Make the bird jump 
    jump: function () {
        // Add a vertical velocity to the bird
        this.bird.body.velocity.y = -350;
    },

    // Restart the game
    restart_game: function () {
        // Start the 'main' state, which restarts the game
        this.game.time.events.remove(this.timer);
        this.game.state.start('main');
        gameStarted=false;
    },

    add_one_pipe: function (x, y) {
        // Get the first dead pipe of our group
        var pipe = this.pipes.getFirstDead();

        // Set the new position of the pipe
        pipe.reset(x, y);

        // Add velocity to the pipe to make it move left
        pipe.body.velocity.x = -200;

        // Kill the pipe when it's no longer visible 
        pipe.outOfBoundsKill = true;
    },
    add_row_of_pipes: function () {
        this.score += 1;
        this.label_score.content = this.score;
        var hole = Math.floor(Math.random() * 5) + 1;

        for (var i = 0; i < 8; i++)
        if (i != hole && i != hole + 1) this.add_one_pipe(400, i * 60 + 10);
    },
};



// Add and start the 'main' state to start the game
game.state.add('main', game_state.main);
game.state.start('main');