Coming from a SQL Server environment before Oracle I can understand the confusion. Oracle does not use a BEGIN TRANSACTION. Instead, a transaction is implicitly started for you.
Thus, I believe in your case SET TRANSACTION NAME is not what you are looking to do, see SET TRANSACTION.
I recommend removing the rollback code from your package and placing it in C#. It should be the responsibility of the caller to commit. Use a transaction in C# to guarantee the the transaction is committed upon successful execution of the package(s).
Ideally, your package structure should look more like this.
declare
ex_custom EXCEPTION;
PRAGMA EXCEPTION_INIT( ex_custom, -20001 );
begin
--Save initial program record
SaveNewRecord(variableName, seq_id);
IF (seq_id != 0) THEN
--If saved successfully, seq_id represents record ID
package_class.secondarySaveMethod(variableName, seq_id);
second_package_class.anotherSaveMethod(variableName, seq_id);
ELSE
-- seq_id invalid throw an exception
RAISE_APPLICATION_ERROR(-20001,'Custom error')
END IF;
htp.p('Sequence ID: ' || seq_id);
htp.p('Saved the record"' || programName || '" successfully!');
EXCEPTION
WHEN ex_custom THEN
-- if needed we log it
utility.log_exception;
-- Raise it for the client to handle
raise;
END SaveApplicationData;