The update seems to be working correctly with the sample data. The results don't match but that's only because you're adding the order number to the description.

CREATE TABLE OE_LINE_RICH(
order_no   VARCHAR(8) NOT NULL PRIMARY KEY
,line_no   DECIMAL(19, 0) NOT NULL
,extended_desc  VARCHAR(255) NULL
,oe_hdr_uid   INTEGER NOT NULL
,generic_custom_description VARCHAR(255) NULL
);
GO
CREATE TRIGGER TI_OE_LINE_RICH ON OE_LINE_RICH
AFTER INSERT
AS
UPDATE o
SET generic_custom_description = i.order_no +' '+ RTrim(LTrim(SUBSTRING(i.extended_desc, CharIndex('Customer Bin:',i.extended_desc)+13,255)))
FROM OE_LINE_RICH o
INNER JOIN INSERTED i
  ON o.order_no = i.order_no
  AND o.line_no = i.line_no
WHERE o.extended_desc like '%Customer Bin:%'
GO
INSERT INTO OE_LINE_RICH(order_no,line_no,extended_desc,oe_hdr_uid,generic_custom_description)
VALUES
   (1000001,1,'MyExtendedDesc Customer Bin: Some Bin Info',2,NULL)
   ,(1000002,3,'MyExtendedDesc Customer Bin: Some Bin Info',3,NULL)
   ,(1000003,12,'MyExtendedDesc',4,NULL)
   ,(1000004,13,'MyExtendedDesc',5,NULL)
   ,(1000005,14,'MyExtendedDesc',6,NULL)
   ,(1000006,15,'MyExtendedDesc',7,NULL)
   ,(1000007,16,'MyExtendedDesc Customer Bin: Some more Bin Info',8,NULL)
   ,(1000008,17,'MyExtendedDesc',9,NULL)
   ,(1000009,18,'MyExtendedDesc',10,NULL)
   ,(1000012,2,'MyExtendedDesc Customer Bin: Some Bin Info',3,NULL)
   ,(1000019,29,'MyExtendedDesc Customer Bin: Even More Bin Info',10,NULL)
   ,(1000622,11,'MyExtendedDesc',3,NULL);
GO
SELECT *
FROM OE_LINE_RICH
GO
DROP TABLE OE_LINE_RICH;