Again, anyone can use an SIC under 135, if they have an approved training program.
The particulars of your question have been spelled out, over and over, throughout the length of this thread, and reading them first would probably be a better choice than repeating the same answered questions over and over.
The regulation, and the operations specifications covering the answer to your question has been provided line by line.
You failed to cite the kind of operation applicable to your operation. If you were in an all cargo operation lacking a requirement for a SIC under Part 135 (the opspecs don't require a SIC for an all cargo operation, as previously explained)...the no, you had no grounds to log the time. If you were a passenger operation, then yes, you did, depending on the authorizations extended your operation, and the type and scope of the operation itself. Again, all previously explained.
Then again, you're not really asking a question, are you? You're arguing a point you don't understand. Go back and read the thread, digest it a little, and you might learn something.
This company also had SIC's who would pay for training.
That would put you in the same pathetic society as 8inman and time builder, then.