Updating a view on multiple joined tables
General view updatability
The key part of the CREATE VIEW (Transact-SQL) documentation is:

Generally, the Database Engine must be able to unambiguously trace modifications from the view definition to one base table.

Note that even if the view is technically updatable, it may not be actually updatable in practice, due to limitations of the query processor's reasoning. That is the subtlety behind the phrase, &quot;...the Database Engine must be able to...&quot;
The easiest way to be sure that a view is actually updatable is to request a pre-execution (&quot;estimated&quot;) plan for the update query. If you get an error, either the view is not logically updatable, or the query processor can't tell that it is.
Requesting a &quot;estimated&quot; plan does not involve executing the query, naturally. The plan shown will also show you how much of the view definition the query optimizer was able to remove (because it is redundant). Typically, it does a good job with this, so the update view plan may look very similar to a plan for a simple update to the single affected base table.
Specific example

Can I use the ClaimStatusName column of the linked table dimClaimStatus to update the main table that is referenced via foreign-key? [...] I want to change fiClaimStatus in tabData.

Not using the query you posted:
update claims 
set status='Awaiting Auth.'
where status = 'Approved' 

This changes the base table column associated with the view's exposed column name status. From the view definition, that is the alias for column  ClaimStatusName in table dimClaimStatus.
The execution plan shows that dimClaimStatus is the table updated through the view:

If you want to update fiClaimStatus, that is the column you need to specify in the update statement. If that involves a lookup, chances are you can't use the view directly, as you originally wanted, but you could write something like:
update claims 
set fiClaimStatus =
(
    select CS.idClaimStatus
    from dbo.dimClaimStatus AS CS
    where CS.ClaimStatusName = 'Awaiting auth.'
)
where status = 'Approved';`