Understanding what Fault, Error and Failure mean

Consider the following class:

class Xyz {
    public int count;

    public void numZero (int[] x) {
        // Effects: if x == null throw NullPointerException
        // else return the number of occurrences of 0 in x
        int count = 0;
        for (int i = 1; i < x.length; i++) //we have a bug here
            {
            if (x[i] == 0)
            {
                count++;
            }
        }

        this.count = count;
    }
}

I'm trying to wrap my head about what Fault, Error and Failure really mean.

Fault

From what I've come to understand, a Fault in this context would be a flaw in the code's written logic. So in this case the Fault would be the fact that the code instructs the computer to start iterating over all elements of v with a start index of 1 instead of the expected 0.

Error

When running the above method, we always get an Error but in once instance (when v.length == 0), as what we really want is to iterate over all elements of x, but since we're starting with i = 1, that is not really happening.

With an empty vector as input, as we don't enter the for loop, so our incorrect code isn't run, meaning that the Error doesn't happen, and everything happens as should in theory.

Failure

Since our code has a Fault that in execution-time will almost always manifest in a Error, we only have a Failure when we effectively see the incorrect output.

Assuming that an Error effectively happened in my program, we only have a Failure if it is in some way visible to the outside world. That is, had I private int count; instead of public int count; I'd never ever have an Error in my class (of course it'd be the most useless class ever!). Is this right?

Is everything I said correct or am I erring in something?

Thanks