- a hole has been drilled or cut through the cover;
- there is a saw cut through the spine of the cover (this was even done to the jewel cases - and back inlays - of CDs);
- one or more corners of the cover have been cut off;
- a hole has been drilled through the labels of the record (especially in the case of US singles);
- there is a stamp on the cover (German "R" stamps).

None of these things indicate a rare record, a promo, or anything like this. On the contrary, these damages show that the record is an overstock copy that couldn't be sold at the regular list price. Here's the reason:

A newly released record is sold to customers at a relatively high price. Which is no wonder, since record companies want to maximise their profits. This means that retail stores also have to pay a relatively high list price for the records (let's call this "price 1"). And, of course, the companies must provide a certain number of records because otherwise customers would have problems in obtaining a copy of that hot new record. But what happens if the record isn't a success and all these copies can't be sold?
In many countries, including the USA, retailers had (and possibly have - I can't say) the right to return such copies to the wholesaler and, ultimately, the record company, and reclaim the original wholesale price ("price 1"). There were two things that the record companies could do with such records: either they could destroy them, and recycle the vinyl (which often led to poor quality pressings made from that stuff - especially in the 1960s and 1970s), or they could try to sell them again. In the latter case the records were offered to retail shops at a much (!) lower wholesale price ("price 2"), so that they could also be sold at a very low retail price. Such records were sold to retailers without the right to return them to the wholesalers (and, ultimately, the record companies).
So far, so good. But who could prevent a tricky record store owner from returning a "price 2" copy and demanding the original "price 1" wholesale price? The only way to avoid this was to mark the overstock records in some unique way. The ways they found are listed above.
I'm not sure if this is the entire story. In the early 1980s, for example, Italian cut-out copies of Iggy Pop's Soldier album were available in heaps in Germany. I can't imagine that the Italians expected the Iggy album to be the next chart-topper. So I think that second pressings were sometimes made with the explicit intention to sell them as cut-outs. In this way you could make at least some money with a record the sales of which were below those expected by the record company.
Be that as it may, in any case a cut-out does not indicate anything special, but either an overstock copy or - possibly - a cheap second pressing!