Multiple databases, one backup file
It is well explained in Paul Randal's article as you mentioned in the question.

Let me walk you through an example.

Server 1 I am taking backup of 3 databases into one file.

BACKUP DATABASE [itdba] TO  DISK = N'D:\multipleDatabaseBackup.bak' WITH NOFORMAT, NOINIT,NAME = N'itdba_backup'
GO
BACKUP DATABASE [itdbatest] TO  DISK = N'D:\multipleDatabaseBackup.bak' WITH NOFORMAT, NOINIT, NAME = N'itdbatest_backup'
GO
BACKUP DATABASE [itdbatest2] TO  DISK = N'D:\multipleDatabaseBackup.bak' WITH NOFORMAT, NOINIT, NAME = N'itdbatest2_backup'
GO


Now if a copy the backup to Server 2 as you mentioned restoring itdba database is like any other restore but you are having trouble restoring other 2 databases.

If you run below tsql 

RESTORE HEADERONLY FROM DISK = N'D:\multipleDatabaseBackup.bak'


you will see the position of backups corresponding to databases.  There are many other columns but I am showing few only.



Take a note of itdbatest database with position 2. I can restore this to another server which does not have the same drive/folder structure as Server 1 using MOVE.

itdbatest database file name and location in server 1.



I copied the backup to Server 2 before restoring.

RESTORE DATABASE itdbatest FROM DISK = N'C:\Program Files\Microsoft SQL Server\MSSQL14.SQL2017\MSSQL\Backup\multipleDatabaseBackup.bak'  
WITH REPLACE, FILE = 2,
MOVE 'ITDBATest' TO 'C:\Program Files\Microsoft SQL Server\MSSQL14.SQL2017\MSSQL\DATA\ITDBATest.mdf',
MOVE 'ITDBATest_log' TO 'C:\Program Files\Microsoft SQL Server\MSSQL14.SQL2017\MSSQL\DATA\ITDBATest_log.ldf'
GO