These cuts were indeed made when the notes were turned in for redemption, which meant gold back in those days. The Republic of Texas did everything they could to default on these obligations, and avoided redeeming them. As a result, these sold on the market for just a few cents on the dollar, to any speculator who wanted to bet on them being paid eventually. When Texas joined the Union, the USA figured that it was not acceptable to default on obligations of what would now be part of the Union, and the money (gold!) was found to finally redeem them, and the speculators came out very well in the end.
Nearly all of these are cancelled, but not all the cancellations had a piece cut out like that, the nicest ones just had the cuts in the note, but not into multiple parts, or pieces removed, etc. A really nice one is coming up for sale in a couple weeks, this shows the full note design.
https://currency.ha.com/itm/obsoletes-by-state/texas/austin-tx-republic-of-texas-10-jan-1-1840-cr-a5-medlar-25-pmg-choice-about-unc-58-epq/a/3599-20137.s