Config
Here's a more extended example of what you can do in your environment in order to say to use mysql but then connect to a specific database created within MySQL:
DB_DEFAULT=mysql
DB_DATABASE=some_database_name
DB_USERNAME=root
DB_PASSWORD=secret
The DB_DEFAULT variable is saying to use the mysql connection type. The DB_DATABASE variable says to connect to a database within MySQL named "some_database_name".
Migration
To make matters a tad more confusing within mysql, there is a database named "mysql". With your setup, your migration has accidentally added new tables (users, password_reset) to the mysql database that's within you mysql server.
This database normally has meta data such as users and access, so mixing in application-related data there is not recommended.
Instead, we usually create a new database per application within MySQL. This can be done with this one liner (which will ask you for your password)
mysql -u root -p -e "create database some_database_name;"
Viewing MySQL Data
phpmyadmin is definitely one option to see the data in your database, but you'll need to install it, confgure it, etc. If you're using the command line, you can also check out the database manually there.
Here are some commands you can try:
mysql -u root -p
> # type in your password here, looks like it's "root"
# Now you get a mysql prompt
Then you can type some mysql commands/query:
show databases;
use mysql;
show tables;
select * from users; # assuming your table is still there
create database some_database_name; -- creates a new database table
If you're using Homestead or a virtual machine, you should be able to use something like SequelPro to view the database as well. How you connect a desktop client application to your mysql database depends on where the database is and how you current log into it.
mysql- that's one of the core databases and the MySQL daemon uses it for statistics and configuration. You shouldn't create tables in that database. You can use a tool like phpmyadmin, although it's a total and utter crap (it's fine for learning purposes, and it's also fine for malicious people to hack you at some point, use at your own risk). There are alternatives, proper GUI programs that don't depend on a web server - such as SQLYog and MySQL Workbench.