There are 2 distinct versions of Microsoft Teams.

Business MS Teams

This is also known as "Teams for Work or School". This is the version of MS Teams every company uses. Meeting links will have the form https://teams.microsoft.com/....

You can sign up for a free Business MS Teams account by following these instructions.

Personal MS Teams

This is also known as "Teams for Home or Small Business". Meeting links will have the form https://teams.live.com/....

If you create a MS Teams meeting with your personal Outlook or Microsoft account, the meeting will be considered a Personal MS Teams meeting.