Updating a field in a Geodatabase

I wanted to update in ArcGIS a point feature class with values representing polygons ID's in which the points are located:

I could do this by means of "Join Data" tool based on spatial location (ArcMap > Table of content > local menu > Joins and Relates > Join...). But this tool creates a new feature class which I do not need.

I could use "Spatial Join" tool from the ToolBox but then also a new feature class  is created ("The output must be unique from the input.").

Maybe there's a way in ArcGIS to just update like I wanted but I couldn't find it (some people say there isn't - see:

http://gis.stackexchange.com/questions/58243/assigning-value-to-a-field-on-a-layer-based-on-spatial-query-with-another-layer).

So I run SQL Server client and typed:

UPDATE Geobaza_08.dbo.PUNKT_ODWIERT

SET OBSZAR_ID = ob.Obszar_ID

FROM Geobaza_08.dbo.PUNKT_ODWIERT po, Geobaza_08.dbo.OBSZAR_BADAN ob

WHERE po.SHAPE.STIntersects(ob.SHAPE) = 1

;

I refreshed the table and I got what I needed:

I tried to make this in ArcGIS without using pure SQL. So I used Python and Arcpy (there should be an edit session opened before executing the following script):

#create update cursor 

uc = arcpy.da.UpdateCursor("PKTO",("OBSZAR_ID","SHAPE@","OBJECTID")) 

#iterate on points

for pkt in uc: 

     #select a polygon containing the point

     arcpy.SelectLayerByLocation_management("OBSZ","CONTAINS",pkt[1]) 

     #create select cursor from the selection

     obs = arcpy.SearchCursor("OBSZ")   

     #iterate on the select cursor 

     for ob in obs: 

          #write the selected polygon id into point field

          pkt[0] = ob.getValue("Obszar_ID")

          #update row in update cursor

          uc.updateRow(pkt)

Hmm... I rather prefer simple SQL update in the DBMS.