Some of this will not align with what NPCs said during the game. The NPCs did not have a full understanding of all of the mechanics of time travel, and some of the conclusions here could not be made in-universe (e.g. whether or not a timeline continues after you leave it cannot be determined in-universe).
A Chronos machine is a device capable of making one end of a bridge through spacetime. This “bridge” is best described as a complex resonance in a higher-dimensional space. By tuning this resonance, a bridge can be created between a Chronos machine and any previous activation of a different machine.
Each Chronos machine can only travel to a given activation once. In general, it is difficult for multiple machines to travel to the same activation (e.g. if machine A is activated in year 1, and machine B travels from year 2 to machine A, then machine C would have to be very precisely calibrated to travel to A). The usual failure mode for this is the instability of the later machine.
Travel back in a Chronos machine creates a new timeline. If nothing is changed during the time travel, then this new timeline may be indistinguishable from the old one. If things change, then there may be notable differences. The old timeline ends at the point where any time travellers would return.
A Chronos machine has a certain level of causal immunity - it will appear in the new timeline at a time and place equivalent to where it was in the original timeline. This causal immunity can be expanded to a significant radius (around the size of a space station or largish asteroid). The Prometheus had causal immunity for the whole station, plus a small region of space around it. The Ananke had causal immunity only for the machine itself.
As an example, consider a future Chronos machine in the year 3100, travelling back to a machine in the year 3000. Timelines will be indicated as 1.3100 for the year 3100 in timeline 1, etc.
Any time after 1.3100 in the first timeline does not exist - that timeline ended when the travellers returned to the future Chronos machine. That's the simple version. Things get more complicated when you consider more tangled travel.
There are now two copies of the diamond, since it has been successfully stolen twice. The alarms that were raised in 2.3001 have been prevented, by ensuring that the travellers who went back to 3.3001 had no reason to trigger them (because the diamond had already been stolen). The versions of the travellers from 3.3001 vanished from existence when the travelled back forwards, because they would have appeared within the causal protection of the future Chronos machine - which would have violated the causal protection of that machine.
If an creature with a mind (generally, an animal) would die while it is travelling back in time, several things happen:
A couple of interesting consequences of this are that an Enhanced person (at the time of the Prometheus) will remember up to the moment of death, but their Opti will only record up to the point they are injured - as the Opti does not have a mind, and so its physical state reverts to the point of injury.
There is no good scientific explanation for this. While there are some signals in the Chronos machine which could be taken to indicate certain properties of the mind which would result in this, they do not actually provide much useful insight into the nature of the mind.
In the CHRONOS universe, minds are a fundamentally nonphysical thing. While a mind can interact with the physical world by controlling the body, it is not made of the same stuff as the body. During time travel, the mind is separated from the body and sent back in time.
This is largely a philosophical point rather than a scientific one, but does have some scientific and practical effects. Notably, true AI comes about because of the possibility of inserting a mind into a machine. AIs in the CHRONOS universe are minds residing in bodies made of code, rather than simply being created by giving sufficient depth to the coding.