SQL Server - Select all polygons nested within a larger polygon

In theory the queries that you have done should return the polygons you said haven't been returned. That makes me suspect that you might be encountering floating point error issues that SQL Server has with it spatial data types. Hence my comment about buffering the bounding polygon with a minimal amount.
So something like the following should get the results you want.

SELECT a.bg10 
FROM gis.usa_10_block_group a
    JOIN gis.usa_10_mkt_definition b
        ON a.shape.STWithin(b.shape.STBuffer(0.0001)) = 1
WHERE b.mktname = 'Loop'

Here's a quick example of the expected behaviour of a few of the spatial methods.

SELECT Geometry::STGeomFromText(WKT,0), Description
    , Geometry::STGeomFromText('POLYGON((0 0, 100 0, 100 100, 0 100, 0 0))',0).STIntersects(Geometry::STGeomFromText(WKT,0)) Intersects
    , Geometry::STGeomFromText('POLYGON((0 0, 100 0, 100 100, 0 100, 0 0))',0).STContains(Geometry::STGeomFromText(WKT,0)) Contained
    , Geometry::STGeomFromText('POLYGON((0 0, 100 0, 100 100, 0 100, 0 0))',0).STOverlaps(Geometry::STGeomFromText(WKT,0)) Overlaps
    , Geometry::STGeomFromText('POLYGON((0 0, 100 0, 100 100, 0 100, 0 0))',0).STTouches(Geometry::STGeomFromText(WKT,0)) Touches
FROM (VALUES
    ('POLYGON((0 0, 20 0, 20 20, 0 20, 0 0))'            ,'Interior corner')
    ,('POLYGON((90 90, 100 90, 100 100, 90 100, 90 90))' ,'Interior corner')
    ,('POLYGON((20 20, 40 20, 40 40, 20 40, 20 20))'     ,'Interior')
    ,('POLYGON((50 0, 70 0, 70 20, 50 20, 50 0))'        ,'Interior edge')
    ,('POLYGON((50 80, 70 80, 70 100, 50 100, 50 80))'   ,'Interior edge')
    ,('POLYGON((80 50, 100 50, 100 70, 80 70, 80 50))'   ,'Interior edge')
    ,('POLYGON((90 0, 110 0, 110 20, 90 20, 90 0))'      ,'Overlap')
    ,('POLYGON((100 50, 120 50, 120 70, 100 70, 100 50))','Exterior edge')
    )P(WKT,Description)
UNION ALL 
SELECT Geometry::STGeomFromText('POLYGON((0 0, 100 0, 100 100, 0 100, 0 0))',0),'Bounding Area',null,null,null,null

Results

Description     Intersects Contained Overlaps Touches
--------------- ---------- --------- -------- -------
Interior corner 1          1         0        0
Interior corner 1          1         0        0
Interior        1          1         0        0
Interior edge   1          1         0        0
Interior edge   1          1         0        0
Interior edge   1          1         0        0
Overlap         1          0         1        0
Exterior edge   1          0         0        1
Bounding Area   NULL       NULL      NULL     NULL

The intersection query should look like this (assuming you want all records back from 'a'):

select a.* --get all columns from table 'a'
from gis.usa_10_block_group a
join gis.usa_10_mkt_definition b
on a.shape.STIntersects(b.shape) = 1
where b.mktname = 'Loop'

If you want just the areas of a that intersect b (ie. clipping a to b), you then add the STIntersection

select a.bg10
, a.STIntersection(b.geom) --clipped geometry from a against b
    from gis.usa_10_block_group a
    join gis.usa_10_mkt_definition b
    on a.shape.STIntersects(b.shape) = 1
    where b.mktname = 'Loop'

But this doesn't get you the polygons that are within b just yet...

This type of polygon-in-polygon is very testy with the boundaries and their conincidence - in order to be 'Within', the boundaries of a cannot be conincident with the boundaries of b - same goes for 'Contains'.

By these definitions, how many of your polygons in a are actually within b...?

So do you want to buffer b before you select polygons in a that are within? Or do a negative buffer on a?

Not sure what the exact answer here is...