How can I safely fall in artistic gymnastics?

This is something you should speak to your coach about. In many ways you have proabably already experienced some traning to fall, without perhaps realising it. For example, this is part of the reason for learning forwards and backwards rolls and why they are worked on to the point they become a natural movement. Rolling is a safer get-out for a lot of falls. When you learn a forwards roll there is that focus on not putting the top of your head down but tucking your chin to your chest and trying to get your shoulders/upper back on the floor. That's partly because you can injure your neck by doing a forwards roll wrongly (though this is more of a problem with young children because their heads are bigger relative to their bodies) but also because this is what you would try to do in a fall. When you learn handstand forwards roll out you are very delibirately learning how to get out of falling on your head. You learn not to let your arms collapse but to slow your descent towards the floor, then roll. I replied to a girl on here a while ago who had freaked herself out by not getting her hands on properly in a handspring vault and crashing out headfirst. She managed to get her hands down and handstand forwards roll out which saved her neck but she felt that she had only not been seriously injured by luck. I pointed out that that wasn't the case. Her training allowed her to save herself.

Rolling isn't always the answer. Gymnasts sometimes practise other fall techniques like falling from feet into front support or jumping up and landing on their front or back on thick crash mats. Things that you want to avoid are landing on your head, landing badly and heavily on one foot (better to fall over completely, especially if you are twisting), landing on your knees (no bendy ankle and knees in play to protect your spine - but landing squarely on hands and knees with your back horizontal is an option,) and landing arched/hyperextended. This is why a lot of the hyperextended poses and balances that apear in rhythmic gymnastics aren't displayed in artistic. Things like forearm stands, chin stand, scorpion pose (not the cheer one the one that's like an extreme yoga bow pose) train hyperextension into your muscle memory, so they don't mix so great with tumbling, beam, bars and valut. Sometimes you can get out of landing hyperextended by twisting.

Of course when you're falling you don't have time for an intellectual assesment of the situation, you act on instinct. That's why training basics and shapes is so important, and why you need to build your skill level gradually so that your sense of where your body is in space and your awareness of how you can move it build along with your skill level.