Linked Server SQL 2016: Can’t update transaction in remote server

There are two major problems that stood out for me:

1. Your execute statement is sending the literal strings (e.g. '@val1') within the query to Instance2 because they're all contained within a string. In order to send the values, you would need to change that part of the execute statement to read:

   Execute ('Update C set col1 = ' + @val1 + ', col2 = ' + @val2
            + ' where col2 = ' + @ID) as user1 at Instance2
   

   Note that the code above assumes both @val1 and @val2 are types which do not need to have quotes around them, otherwise you'd use the following:

   Execute ('Update C set col1 = ''' + @val1 + ''', col2 = ''' + @val2
            + ''' where col2 = ''' + @ID + ''') as user1 at Instance2
   

2. Your query is not benefiting from any potential preoptimization on Instance2. Therefore, I would suggest creating a stored procedure on Instance2 so that you could benefit from pre-runtime optimization of the query, and also use all the optimization hints that you might want to place in the query. So on Instance2, you could create a procedure like this (note again that I have assumed integer datatypes):

   CREATE PROCEDURE user1.UpdateC (@val1 int, @val2 int, @ID int) AS
   BEGIN
       UPDATE C WITH (ROWLOCK) SET col1 = @val1, col2 = @val2 WHERE col2 = @ID
   END
   

   Then, your local script could replace the corrected code in part 1 above with the following (again, assuming integers):

   EXECUTE ('EXECUTE user1.UpdateC ' + @val1 + ', ' + @val2 + ', ' + @ID)
       AS user1 AT Instance2
   

Code in triggers runs inside the same implicit transaction as the insert/update/delete statement. Therefore, it's trying to share the transaction across both servers, which is disallowed by default.

Do you have enough privilege to change settings? If not, then you could consider other options such as inserting into a local table and replicating the data across, or use Service Broker to get another process to do the remote insert.