Maybe this query will work:
SELECT 'My_Linked_Server', database_name,
MAX(backup_finish_date) AS last_db_backup_date,
case backupset.type
when 'D' then 'Database'
when 'I' then 'Differential database'
when 'L' then 'Log'
when 'F' then 'File or filegroup'
when 'G' then 'Differential file'
when 'P' then 'Partial'
when 'Q' then 'Differential partial' end as [Type]
FROM [My_Linked_Server].msdb.dbo.backupmediafamily
INNER JOIN [My_Linked_Server].msdb.dbo.backupset ON backupmediafamily.media_set_id = backupset.media_set_id
inner join [My_Linked_Server].master.sys.databases ON databases.name = backupset.database_name
WHERE state_desc = 'ONLINE'
GROUP BY backupset.database_name, backupset.type
ORDER BY backupset.database_name, backupset.type
You can also put the types and abbreviations to some helper table and use a JOIN.