We do not know if what was described as a "sanity test" actually meets your definition of a sanity test, or was used for some other purpose like a smoke test.
A quick look using Google Scholar finds people using "sanity test" for smoke testing, like
"Typical behavior is to allocate any special resource requirements it needs, map the device into virtual address space, initialize the device and perform a brief “sanity test” to ensure that the device appears to be working correctly" at https://onlinelibrary.wiley.com/doi/pdf/10.1002/%28SICI%2910...
or
"The minimum essential test cases that need to be executed to evaluate the essential functionality are known as Sanity Test Cases" https://uksim.info/icaiet2014/CD/data/7910a048.pdf
or
"Sanity test is a brief run-through of the functionality of the software system to assure that the system works as expected." https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6006830
But it's also used for full testing, including human intervention, like
"There will always be a role for manual testing. For one, it is the only real way to sanity-test your automation itself." - https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&d...
and
"Sanity test cases which checks basic functionality and are run for pre-system acceptance and when product goes thru major change. These test cases deliver a
very high project value to both engineering dept and to customers." https://ijaer.com/admin/upload/06%20Apr_2012_Sheo%20Kumar.pd...
as well this example which distinguishes between shallow-and-wide smoke tests with deep-and-narrow sanity tests.
"a smoke test generally consists of a collection of tests that can be applied to a newly created or repaired computer program. This is a “shallow and wide” approach to the application." while "Sanity testing will be performed whenever cursory testing is sufficient to prove that the system is functioning according to specifications. A sanity test is a narrow regression test that focuses on one or a few areas of functionality. Sanity testing is usually narrow and deep. It will normally include a set of core tests of basic GUI functionality to demonstrate connectivity to the database, application servers, printers, etc." - http://archives.christuniversity.in/disk0/00/00/48/68/01/the...
Going with PC Replacement question, since we don't know the actual goal of what was called "sanity test", we can't ourselves come up with a better name.
Which is why we should go with the OP's assessment that "quick" is an appropriate term for what they are doing. Since you don't know what that is, it doesn't matter if you buy a new term or not.