Logs from Bitcoin Core activity
No, that text is not proof that there were transactions from your wallet. It just means the the software was running and those lines are normal. They just indicate that the received data from peers, but that data is not necessarily related to you.

To check if you have any Bitcoin, you can run the software and see what it tells you. It sounds like you are using an old version of Bitcoin Core, so you should probably upgrade to the latest Bitcoin Core first as it will sync much faster. Because that is Bitcoin Core, you will need to download the entire blockchain which is 180+ GB of data, but you can enable pruning to reduce the disk space used.