[Vuejs]-Two compile cases with Webpack, Vue.js, and Sass

Sorry PlayMa256 & Máté, for being so long before answering your replies.
In the end I found the solution of using two different configurations for my two cases. Webpack allows it through its multi-compiler feature.

So here is what my webpack.config.js now looks like:

module.exports = [ // notice that we are handling an array of configurations here

    // In this first config, I manage the first of my use cases: 
    // Compilation of .scss files into .css files
    {
        name: "css",
        entry: { /* ... */ },
        output: { /* ... */ },
        /* ... */
        module: {
            rules: [
                {
                    test: /\.scss$/,
                    use: [ MiniCssExtractPlugin.loader, 'css-loader', 'postcss-loader', 'sass-loader' ],
                }
            ]
        },
        plugins: [ /* ... */]
    },

    // In this other config, I manage the other of my use cases: 
    // injection of the <style> blocks of my .vue files into the DOM
    {
        name: "main", // name for first configuration
        entry: { /* ... */ },
        output: { /* ... */ },
        /* ... */
        module: {
            rules: [
                // Vue single file components loader
                {
                    test: /\.vue$/,
                    loader: 'vue-loader',
                },

                // Injection of <style> elements into the DOM,
                // for both plain `.css` files and `<style>` blocks in `.vue` files
                {
                    test: /\.css$/,
                    use: [
                      'vue-style-loader',
                      'css-loader'
                    ]
                },

                // Compilation of sass code,
                // (This actually works both for `.css` files and `<style>` blocks in `.vue` files, 
                // but I don't have any `.css` as entry for this config.)
                {
                    test: /\.scss$/,
                    use: [
                        "style-loader", // creates style nodes from JS strings
                        "css-loader", // translates CSS into CommonJS
                        "sass-loader" // compiles Sass to CSS, using Node Sass by default
                    ]
                }
            ]
        },
        plugins: [ /* ... */]
    }
];