How to configure eslint on vue-cli project to work correctly

Question

I have a vue project that build successfully, but when I add in the eslint option in package.json file I receive Parsing error: Unexpected token pointing at a private variable. I have been researching for the last few hours and have been unsuccessful at finding a solution that works.

sites I have visited, and solutions tried

https://stackoverflow.com/a/49129502/4250126

https://github.com/vuejs/eslint-plugin-vue/issues/746

https://stackoverflow.com/a/53609667/4250126

https://dev.to/ferasdawod/setting-up-eslint-for-a-vuejs-project-using-vs-code-i54

and some others...

I just can't seem to figure out why lint is complaining about a variable. screen shot of running vue-cli-service lint from package.json. below I have tried changing public to private or removing the underscore, but it does not seem to help

my package.json

{
  "name": "data-dictionary",
  "version": "0.1.0",
  "private": true,
  "scripts": {
    "serve": "vue-cli-service serve",
    "build": "vue-cli-service build",
    "lint": "vue-cli-service lint"
  },
  "description": "DataDictionary",
  "author": {
    "name": ""
  },
  "dependencies": {
    "@typescript-eslint/parser": "^2.4.0",
    "axios": "^0.19.0",
    "bootstrap-vue": "^2.0.4",
    "buefy": "^0.8.5",
    "eslint-plugin-import": "^2.18.2",
    "vue": "2.5.17",
    "vue-property-decorator": "^8.2.2"
  },
  "devDependencies": {
    "@vue/cli-plugin-babel": "3.0.4",
    "@vue/cli-plugin-eslint": "3.0.4",
    "@vue/cli-plugin-typescript": "^3.12.0",
    "@vue/cli-service": "3.0.4",
    "babel-eslint": "^10.0.3",
    "eslint": "5.6.0",
    "eslint-plugin-vue": "^4.7.1",
    "typescript": "^3.6.4",
    "vue-template-compiler": "2.5.17"
  },
  "eslintConfig": {
    "root": true,
    "env": {
      "node": true
    },
    "extends": [
      "plugin:import/typescript",
      "plugin:vue/essential",
      "eslint:recommended"
    ],
    "rules": {
    },
    "parserOptions": {
      "parser": "babel-eslint",
      "sourceType": "module",
      "ecmaFeatures": {
        "legacyDecorators": true
      },
      "ecmaVersion": 6
    }
  },
  "eslintIgnore": [ "/**/*.d.ts" ],
  "postcss": {
    "plugins": {
      "autoprefixer": {}
    }
  },
  "browserslist": [
    "> 1%",
    "last 2 versions",
    "not ie <= 8"
  ]
}

Any help is appreciated. I am probably doing something completely wrong, but not extremely familiar with using lint with Vue.

UPDATE If I remove the public/private keywords then vue-cli-service lint passes.