My solution

//Random number from 1 to the given number
function ranForYou() {
    var choice;
    do {
        choice = prompt("This machine will create a random integer Number between 1 and the number you choose. So please choose a number first. It should be a NUMBER and bigger than 1.");
    } while(isNaN(parseInt(choice)) || parseInt(choice) <= 1);
    //used Math.ceil() instead of Math.floor(...) + 1
    var randomNumber = Math.ceil(Math.random() * parseInt(choice));
    alert("The random number is " + randomNumber);
}

//Random number between the 2 given numbers
function randomForYou() {
    var choice;
    do {
        choice = prompt("This machine will create a random integer Number between the 2 numbers you choose. So please choose both numbers first. Make sure there are TWO NUMBERS. Separate them with A SPACE please");
        var numbers = choice.split(" ");
    } while(isNaN(numbers[0]) || isNaN(numbers[1]) || numbers[0] === '' || numbers[1] === '' || numbers.length > 2);
    //Math.ceil() keeps the result of "2 3" always 3, so changed to Math.round()
    var randomNumberBase = Math.round(Math.random() * (parseInt(numbers[1]) - parseInt(numbers[0])));
    alert("The random number is " + (randomNumberBase + parseInt(numbers[0])));
}