"We can respectfully agree to disagree"
Actually, we agree on a fair amount.
I don't make a lot of effort because the chance of winning enough to matter and then having somebody else try to claim it's their ticket is awfully close to zero, but some things require almost zero effort. I also play the same numbers for long periods of time. I don't keep all of the old tickets, but I've got a few old tickets (from multiple states) tucked away, including the first (or at least early) time I use a combination. I don't bother with receipts for ticket purchases, but there are plenty of credit card transactions showing that I've bought gas immediately before tickets with my numbers were purchased. There are also email servers for a rather large company that have proof that I knew when and where some of those tickets would be bought, as well as how the buyer would be dressed, a day or two before the purchase was made. FWIW, here in NY you can claim prizes without a ticket, as long as you can provide convincing proof that you own the missing ticket.
I happened to recently be in a state where I could remain anonymous and bought tickets for both PB and MM. I'm sure enough that I can easily establish ownership that I kept my head down in hopes that I wouldn't be easily recognized on the video if it were to be released to the public.
"If they determined that I erased my name in favor of a trust, the name erased would match the name of the owner of the trust"
I'd have thought that the rules about altered tickets is about alterations that attempt to portray a losing ticket as a winner and that any changes to the signature would only cause problems about ownership, but one of the other articles about Jane Doe's ticket reports that the NH lottery "told her she can't white-out her name from the back of the ticket, as any alteration (my emphasis) would invalidate it."
I'm sure (and it sounds like you are, too) that inability to establish legitimate ownership is extremely unlikely even without a signature. The chance that erasing a signature would turn a winning ticket into a worthless souvenir, OTOH, seems significant.