EF Core: how do I configure mySql for Linux, and localDB for Windows?

I have a simple ASP.Net Core/Entity Framework Core project that uses LocalDB. It compiles and runs fine on Windows.

I would like to build and run the same project on Windows and Linux. But LocalDB isn't supported on Linux. So I need to configure the project to use mySql instead - but only for Linux.

Q: How do I configure my project so that I can use LocalDB on Windows, but mySql on Linux?

This is what I've tried so far:

1. Created an empty mySql database and granted access to a mySql user.

2. Created a mySql connection string in appsettings.json:

   {
     "ConnectionStrings": {
     "DefaultConnection": "Server=(LocalDb)\\MSSQLLocalDB;Database=ManageCarDB;Trusted_Connection=True;MultipleActiveResultSets=true",
     "mySqlConnection": "Server=localhost;port=3306;database=ManageCarDb;uid=dotnetuser;password=dotnetuser"
   },...
   <= I've defined two different connection strings: one for LocalDB, one for MySql
   

3. Updated Startup.cs:

   public void ConfigureServices(IServiceCollection services)
   {
       string env = System.Environment.GetEnvironmentVariable("ASPNETCORE_ENVIRONMENT");
       string connectionString;
       if (!string.IsNullOrEmpty(env) && env.Equals("Linux"))
       {
          connectionString = Configuration.GetConnectionString("mySqlConnection");
           services.AddDbContext<ApplicationDbContext>(options =>
               options.UseMySQL(connectionString));
       }
       else
       {
           connectionString = Configuration.GetConnectionString("DefaultConnection");
           services.AddDbContext<ApplicationDbContext>(options =>
               options.UseSqlServer(connectionString));
       }
   

   <= Startup will conditionally call either MySql connection string/MySQL data provider, or default/LocalDB

4. On Linux:

   1. deleted all binaries
   2. dotnet restore
   3. dotnet ef migrations add newMigration -c ApplicationDbContext -v

   <= This all worked OK

5. Tried to update the database:

   dotnet ef database update

   <= ERROR: Table 'ManageCarDb.__EFMigrationsHistory' doesn't exist

Q: Given that I'd like one project for both EF environments, am I taking the correct steps?

Or should I taking a different approach?