Firstly, I'd advise a read of this: Is storing a delimited list in a database column really that bad? (Spoiler, the answer is yes).

Next I'd advise a read of Divided We Stand: The SQL of Relational Division , this covers a wide range of approaches to relational division (what you are trying to do) for each of the scenarios you've covered.

Finally, the approach I would take is to first split your values into rows using STRING_SPLIT then do your equality checks, then roll these back up to the original row level, something like:

DECLARE @Values VARCHAR(1000) = '00001,00002';
WITH SplitInput AS
(
    SELECT  DISTINCT ss.value
    FROM    STRING_SPLIT(@Values, ',') AS ss
)
SELECT sd.GenericCode,
        ContainsAny = CASE WHEN COUNT(si.value) > 0 THEN 1 ELSE 0 END,
        ContainsAll = CASE WHEN COUNT(si.value) = (SELECT COUNT(*) FROM SplitInput) THEN 1 ELSE 0 END,
        ContainsOnly = CASE WHEN COUNT(si.value) = COUNT(*) AND COUNT(si.value) = (SELECT COUNT(*) FROM SplitInput) THEN 1 ELSE 0 END
FROM    #SampleData AS sd
        CROSS APPLY STRING_SPLIT(sd.GenericCode, ',') AS ss
        LEFT JOIN SplitInput AS si
            ON si.value = ss.value
GROUP BY sd.GenericCode;

For a sample data set this returns something like the following for the input above (00001,00002)

• Contains Any = Any of the supplied values match any of the values stored
• Contains All = All of the supplied values match the stored values, does not matter if there are additional stored values
• Contains Only = All of the supplied values are matched with no remainder (i.e. exact match)

Hopefully you can adapt this to suit your different scenarios.

Full Code example

DROP TABLE IF EXISTS #SampleData;
CREATE TABLE #SampleData (GenericCode VARCHAR(100));
INSERT #SampleData(GenericCode)
VALUES ('00001'),('00001,00002'),('00001,00034'), ('00001,00002,00034,00052');
GO
DROP PROCEDURE IF EXISTS #SampleProcedure; -- Using temp proc for simplicity of demo code
GO
CREATE PROCEDURE #SampleProcedure @Values VARCHAR(1000)
AS
BEGIN
    WITH SplitInput AS
    (
        SELECT  DISTINCT ss.value
        FROM    STRING_SPLIT(@Values, ',') AS ss
    )
    SELECT sd.GenericCode,
            ContainsAny = CASE WHEN COUNT(si.value) > 0 THEN 1 ELSE 0 END,
            ContainsAll = CASE WHEN COUNT(si.value) = (SELECT COUNT(*) FROM SplitInput) THEN 1 ELSE 0 END,
            ContainsOnly = CASE WHEN COUNT(si.value) = COUNT(*) AND COUNT(si.value) = (SELECT COUNT(*) FROM SplitInput) THEN 1 ELSE 0 END
    FROM    #SampleData AS sd
            CROSS APPLY STRING_SPLIT(sd.GenericCode, ',') AS ss
            LEFT JOIN SplitInput AS si
                ON si.value = ss.value
    GROUP BY sd.GenericCode;

END
GO
EXECUTE #SampleProcedure @Values = '00001';
EXECUTE #SampleProcedure @Values = '00001,00002';
EXECUTE #SampleProcedure @Values = '00001,00002,00034';
EXECUTE #SampleProcedure @Values = '00001,00002,00034,00052';
EXECUTE #SampleProcedure @Values = '00035,00055';