Quite a few Counter-stamps are commercial. Many copper pennies were stamped with "Lloyds" and used as tokens for their paper. The Chinese used Chop marks to confirm authenticity or claim ownership.
Some were also governmental - indicating a re-valuation or the use in subsidiary or revolutionary organisations. Many early Caribbean coins are counter stamped versions of other country's money.
Engraving tends to be more personal, individual and so difficult to track down. When I first started collecting again in the early 2000s I picked up a cartwheel penny at an antiques shop, which, in the terms of the vendor, had been "Vandalised as someone's written all over the back" As a result it was cheap. The wording was something like "When this you see, think of me, J Bond 1827" - it was a transportation token created by or for someone about to be shipped to the colonies! I put it on Ebay, and it sold for a tidy sum, going to Australia. Sadly I no longer have a picture.
Here is a Brazilian 20 Reis counter stamped by the government to revalue as 40 Reis.