What is the preferred / common way to separate "access-tokens" in a database design
The answer is neither.  Since a user could have multiple tokens, the primary key is not the same and it's meaning within the context of the application is not the same as the tokens used for email verification/password reset.
So:

You have a table for session_token.  If you will search by token to validate a user then the primary key is token.  If you will primarily search by user the primary key will be (user_id,token) with an unique constraint on token.
You have a table for user_token or something similar.  The primary key is (user_id,token_type).  You may or may not need an alternate key on token depending on how things are used.  This is preferable to separate tables as you can add a new type without creating a new table.  You will need a table token_type that provide integrity to that column.  Use values that are easy to understand.

Do not slap an row identifier on each table (id) as:

It does not provide uniqueness
It is unnecessary to replace the existing primary key
It is counterproductive as it will never be used to access either table.