Package issues with babel - reactjs

Question

Community, just started with React and I've created the simplest stuff ever. Does it work? Of course... not.
I've been googling and tried to make those adjustments described. But still, my killer application won't run and I seem to end up with two different error messages depending on the actions i take:
"Babel Plugin/Preset files are not allowed to export objects, only functions"
"Uncaught ReferenceError: process is not defined"
I honestly don't know how to proceed, I've tried to upgrade to the latest "#babel/...", tried to downgrade to earlier babel versions, but React is playing hide and seek with me.
// app.js
import React from 'react';
import ReactDOM from 'react-dom';
const text = <p>Hello world</p>
ReactDOM.render(text, document.getElementById('app'));
// babel.rc
{
"presets": [
"env",
"react"
]
}
// webpack.config.js
const path = require('path');
module.exports = {
entry: './src/app.js',
output: {
path: path.join(__dirname, 'public'),
filename: 'bundle.js'
},
mode: 'none',
module: {
rules: [{
loader: 'babel-loader',
test: /\.js$/,
exclude: /node_modules/
}]
}
};
// package.json
{
"name": "killer-app",
"version": "1.0.0",
"main": "index.js",
"author": "",
"license": "",
"scripts": {
"serve": "live-server public/",
"build": "webpack --watch",
"build-babel": "babel src/app.js --out-file=public/scripts/app.js --presets=env,react --watch"
},
"dependencies": {
"babel-cli": "^6.26.0",
"babel-core": "6.25.0",
"babel-loader": "7.1.1",
"babel-preset-env": "1.5.2",
"babel-preset-react": "6.24.1",
"live-server": "^1.2.2",
"react": "^18.2.0",
"react-dom": "^18.2.0",
"validator": "^13.7.0",
"webpack": "^5.73.0"
},
"devDependencies": {
"webpack-cli": "^4.10.0"
}
}
As a side-note, the script "yarn run build-babel" doesn't work either. This script worked before I installed Webpack.
Any advice is appreciated! :)