Sql-server – Why is the update affecting twice the number of records than expected

The concept you want to understand is called "ownership chaining"

As long as the owner of the object and database remain the same, SQL Server will not re-evaluate the permissions of a user after the 1st object in the chain.

In your first example you denied USER1 access to TABLE1 but allowed access to VW_TABLE1. Since USER1 has access to VW_TABLE1 and the ownership chain has not been broken, they can see TABLE1 because permissions are not re-evaluated inside the VIEW.

In your second example I'm assuming the SEC schema must have a different owner which causes SQL Server to re-evaluate permissions when inside the VW_TABLE1 view.

In short, to fix this problem make sure all objects have the same owner

Not seeing the data you have makes this a bit harder but I managed to reproduce what you see if you see with this:

insert into RunDetails(RunDetailID, JobDetailID) values(448, 1)
insert into MRDFStorage(RunDetailID, SortField) values(448, 'S1')
insert into MRDFStorage(RunDetailID, SortField) values(448, 'S1')
insert into MRDFStorage(RunDetailID, SortField) values(448, 'S1')
insert into BallotStyles(JobDetailID, Style) values(1, 'S1')
insert into BallotStyles(JobDetailID, Style) values(1, 'S1')

The key factor here is the two rows in BallotStyles.

You are joining to BallotStyles in the inner most query and with data like above you will get duplicate rows for each row in MRDFStorage. Since you are not using any of the columns in BallotStyles you are only checking for existence of rows and that can be done with a exists clause instead and that will of course not create the duplicate rows.

UPDATE
    [MRDFStorage]
SET
    [Seq9] = [OrderedRecords].[NewSeq9]
FROM
    (
        SELECT
            MRDFStorageID,
            [Records].[SortField] + RIGHT('0000' + CAST(ROW_NUMBER() OVER(PARTITION BY [Records].[SortField] ORDER BY [Records].[RunDetailID], [Records].[SortField], [Records].[PieceID]) AS VARCHAR), 4) NewSeq9
        FROM
        (
            SELECT
                M.[MRDFStorageID], 
                M.[RunDetailID], 
                M.[SortField], 
                M.[PieceID], 
                M.[BallotType]
            FROM
                [MRDFStorage] as M
-- Remove this join
--                    JOIN [BallotStyles] ON M.[SortField] = [BallotStyles].[Style] and [BallotStyles].[JobDetailID] = @JobDetailID
            WHERE
                [RunDetailID] IN (SELECT [RunDetailID] FROM [RunDetails] WHERE [JobDetailID] = @JobDetailID) and
-- Add this exists check instead of the join
                EXISTS (
                       SELECT *
                       FROM BallotStyles AS BS
                       WHERE M.SortField = BS.Style and
                             BS.JobDetailID = @JobDetailID
                       )
        ) Records
    ) OrderedRecords 
        JOIN MRDFStorage ON [OrderedRecords].[MRDFStorageID] = [MRDFStorage].[MRDFStorageID]
WHERE
    [MRDFStorage].[RunDetailID] = @RunDetailID