How can I change the EditText text without triggering the Text Watcher?

Short answer

You can check which View currently has the focus to distinguish between user and program triggered events.

EditText myEditText = (EditText) findViewById(R.id.myEditText);

myEditText.addTextChangedListener(new TextWatcher() {

    @Override
    public void onTextChanged(CharSequence s, int start, int before, int count) {
        if (myEditText.hasFocus()) {
            // is only executed if the EditText was directly changed by the user
        }
    }

    //...
});

Long answer

As an addition to the short answer:
In case myEditText already has the focus when you programmatically change the text you should call clearFocus(), then you call setText(...) and after you you re-request the focus. It would be a good idea to put that in a utility function:

void updateText(EditText editText, String text) {
    boolean focussed = editText.hasFocus();
    if (focussed) {
        editText.clearFocus();
    }
    editText.setText(text);
    if (focussed) {
        editText.requestFocus();
    }
}

For Kotlin:

Since Kotlin supports extension functions your utility function could look like this:

fun EditText.updateText(text: String) {
    val focussed = hasFocus()
    if (focussed) {
        clearFocus()
    }
    setText(text)
    if (focussed) {
        requestFocus()
    }
}