Most police departments will enter a mark you put on the saw in a police report and it will stay with that agency. The National registries (NCIC) use make, model and serial # for identification of stolen items. The rest is fine if the reporting agency has more information, the more info you have the better. But for the cop on the street stopping a suspected thief, burglar, etc, he is going to run the items make, model and serial # to determine if it is stolen to recover. When we do search warrants on suspected thiefs, burglars houses, that's what we run, make, model and serial #. If we see an operation ID number we will run that also. For police to "take" an item from a suspected thief/burglar we have to have proof that it's stolen. Just marking an item doesn't prove that. The computer (NCIC) has to have the make, model and serial # entered as a stolen item for us to "recover" the item. If you put an Owner Applied Number (OAN) hidden on the item, it may not be found by the police during a stop/warrant. Most officers are trained to run the make, model and serial number through NCIC for a hit verification of stolen.
As for proof of ownership, that's a different story. If someone calls in a stolen chainsaw, it is presumed that they are telling the truth. If the item is recovered and the current "owner" of the item disputes that, then it's a whole different story. Maybe the original reporter is a swindler or fraud person. Proof of ownership is different and a drivers license number on the item does not prove ownership. This is where you will need your receipt with serial number on it, to prove ownership. The person with the drivers license number on the item may have stolen it and just put their DL number on it. No proof with that.
Most Pawn Systems are local, to stop the local thief from continually pawning items, or tracking people who do lots of pawning of specific type items. All licensed pawn shops (at least here in the Twin Cities) are required to enter make, model and serial number of items brought into their stores for pawning.
Problem with the internet and ebay today, is that many states do things differently. They have different computer databases, but they don't talk well with each other. The one main system I've seen used all over the US is NCIC which mainly uses Make, Model and Serial #.
Drivers license numbers don't always work, had people try that. If you move and don't notify DMV of your new address, or move out of state, they can be useless. The only time I've seen "markings" make a difference in a recovery was in jewelry, where the stone had gone to a jeweler and had a flaw map drawn out. These were some pretty valuable jewels though, to justify the cost.
To me, your best bet is to use make, model, model serial and keep receipts. If you choose to put a mark on the saw, great, for ID purposes once it is recovered. But to help recovery, keep the make, model and serial number along with a receipt.
Tom