Are checksums basically toned-down versions of cryptographic hashes? As in: they are supposed to detect errors that occur naturally/randomly as opposed to being designed to prevent a knowledgeable attacker's meticulous engineering feat?
That is one way to look at it. However, hash functions have many purposes. They are also meant to be one-way (an attacker cannot know the preimage without guessing), for which there is no parallel with checksums.
So, essentially they are non-secure versions of cryptographic hashes, one could say? Thus for the same reason, these checksums are "cheaper" to compute than cryptographic hashes? (e.g. CRC32 vs SHA-256)
Due to their different requirements, checksums are not just "worse, but faster hashes". They are meant to prevent particular kinds of errors. Cyclic redundancy check can detect e.g. all 1-2 bit errors in short inputs, as well as some other common classes of errors in typical applications (e.g. bursts errors). This is better than a truncated cryptographic hash of similar length would be able to do.
A cryptographic hash truncated to 32 bits can easily collide with two inputs that differ in only one or two bits, whereas a CRC won't. The CRC is geared towards reliably detecting error patterns that commonly occur in transit, so it will do better on those kinds of errors and worse on others. The short hash does optimally over all inputs, and as a result does worse than CRC on the inputs CRC is good at dealing with.