I work at a shop, put that SN for your 044 into the two places I know to get a date of manufacture, the saw was either never registered or is too old to be in Stihls system. I’m guessing it’s too old.
Stihl has a nice parts lookup feature when you put the SN in it gives you a month and year of manufacture but it only works with saws that have the MS in the model number. Like an MS 440 will pop up but an 0-series saw like an 044 will not because of how old the saw is.
If you post the SN of the 361 I can check it out. All 362’s come up, but an older 361 may or may not, like I said most saws that have the MS come up