I renamed the methods based upon where they're triggered:

<body>
    <div id="root">
        <coupon @applied="rootMethodTriggered"></coupon>
        <span v-show="showText">This text is linked to a variable (showText) in the Vue instance, now visible</span>

    </div>
<script src="https://cdn.jsdelivr.net/npm/vue/dist/vue.js"></script>
<script>
Vue.component("coupon", {
  template: `
  <div>
    <input @blur="localMethodTriggered">
    </div>
            `,
  methods: {
    localMethodTriggered() {
        console.log('this is the local method, triggered on blur');
            this.$emit("applied");
        }
    }
});

new Vue({
    el: '#root',
    data: {
        showText: false,
    },
    methods: {
        rootMethodTriggered(){
            console.log('applied was emitted which is bound to rootMethodTriggered, this is the root method');
            this.showText = true
        }
    }
})
</script>
</body>

Component Communication Example #1: Custom Events

Updated Series Available