Using Extended Event For Tracking Updates To A Table
When using SQL Server Enterprise Edition (or running any edition newer than 2016 SP1) I would recommend going with SQLAudit functionality. This will give you some great granualar information about who is touching your tables, and the commands that are being executed.

For something like this you would want to use a database audit specification along with a server audit (used to define where your audit will write to), and then scope it down to the table that you want to monitor. By then scoping changes to the public role you will capture any and all changes that happen. 

This script should get you there (test in your dev environment, and replace the relevant pieces for the table(s) that you want to monitor).

USE [master]

GO

CREATE SERVER AUDIT [Audit-TblChanges]
TO FILE 
(   FILEPATH = N'C:\SQLAudit'
    ,MAXSIZE = 0 MB
    ,MAX_ROLLOVER_FILES = 2147483647
    ,RESERVE_DISK_SPACE = OFF
)
WITH
(   QUEUE_DELAY = 1000
    ,ON_FAILURE = CONTINUE
)

GO

USE [YourDatabase]
GO

CREATE DATABASE AUDIT SPECIFICATION [DatabaseAuditSpecification-MyTable]
FOR SERVER AUDIT [Audit-TblChanges]
ADD (UPDATE ON OBJECT::[YourTable] BY [public]),
ADD (INSERT ON OBJECT::[YourTable] BY [public])

GO


USE [master]
GO
ALTER SERVER AUDIT [Audit-TblChanges] WITH (STATE = ON);

USE [YourDatabase]
GO
ALTER DATABASE AUDIT SPECIFICATION [DatabaseAuditSpecification-MyTable] WITH (STATE = ON);
GO


You can quickly read the data back using the GUI, or use sys.fn_get_audit_file to query the data directly from SQL Server.