Those articles simply refer to the viability of the virus, i.e. whether it can still infect a host thereafter. (Sometimes this is referred to as "survival" of the virus.)
A unviable virion is not quite the same as it being "not there" at all, although it is so from the perspective of a host that doesn't get infected.
Those viability [in the environment] studies use a quantitative measure of how much of the virus (i.e. how many virions) are still viable after some time on some surface (or even in "the air"), e.g.
SARS-CoV-2 remained viable in aerosols throughout the duration of our experiment (3 hours), with a reduction in infectious titer from 103.5 to 102.7 TCID50 per liter of air. [...]
SARS-CoV-2 was more stable on plastic and stainless steel than on copper and cardboard, and viable virus was detected up to 72 hours after application to these surfaces [...], although the virus titer was greatly reduced (from 103.7 to 100.6 TCID50 per milliliter of medium after 72 hours on plastic and from 103.7 to 100.6 TCID50 per milliliter after 48 hours on stainless steel).
So they are measuring the time to some [chosen] statistical threshold of viability. For reference as to their particular method:
The TCID50 (Median Tissue Culture Infectious Dose) is one of the methods used when verifying viral titer.
TCID50 signifies the concentration at which 50% of the cells are infected when a test tube or well plate upon which cells have been cultured is inoculated with a diluted solution of viral fluid.
Which part of the virus machinery breaks first might depend on the environment. On copper, for example, it has been found that for "common cold" coronaviruses (e.g. HuCoV-229E):
Exposure to copper destroyed the viral genomes and irreversibly affected virus morphology, including disintegration of envelope and dispersal of surface spikes. Cu(I) and Cu(II) moieties were responsible for the inactivation, which was enhanced by reactive oxygen species generation on alloy surfaces, resulting in even faster inactivation than was seen with nonenveloped viruses on copper.
Coronavirus was exposed to metal surfaces and recovered, and the positive-stranded viral RNA genome was extracted and purified. A one-step reverse transcriptase real-time quantitative PCR (RTqPCR) was performed to detect a 139-bp region of ORF1 within nonstructural protein 4 (nsp4). Virus that had been exposed to copper and brass surfaces demonstrated reduced copy numbers of this fragment with increasing contact times. [...] . Comparison of the entire viral genome by agarose gel electrophoresis confirmed that nonspecific fragmentation occurred on copper and brass, with fragments becoming smaller with increasing contact time.
See also related q here on virus survival in small droplets, where a different inactivation mechanism is suspected.
In general, all viruses start to degrade outside their hosts at temperatures above -20C, even in the absence of some more specific environmental stressors (like unfavorable humidity, reactive surfaces like Cu, etc.) In suitable hosts, this temperature-based degradation is more than offset by the virus' multiplication.