How can I check whether a domain is occupier without fear that whois provider will immediately register it?

There are many horror stories when people checked domains, found not occupied, but when came back to register the best one, all of them appeared to be registered by someone else. So, what is the safest way to check domains for being occupied?

Don't use a website. Use the whois command as this should go directly to the registry (not registrar) whois service. It will only refer to the registrar's whois if it is already registered.

whois doesthisreallyactuallyexist.com

Whois Server Version 2.0

Domain names in the .com and .net domains can now be registered
with many different competing registrars. Go to http://www.internic.net
for detailed information.

No match for "DOESTHISREALLYACTUALLYEXIST.COM".
...snip...

You can explicitly set the whois server and query type if you wish:

whois -h whois.verisign-grs.com domain doesthisreallyactuallyexist.com

Whois Server Version 2.0

Domain names in the .com and .net domains can now be registered
with many different competing registrars. Go to http://www.internic.net
for detailed information.

No match for domain "DOESTHISREALLYACTUALLYEXIST.COM"

Go to the source. whois [email protected] or alternatively go to http://www.arin.net and whois from their website. ARIN isn't going to steal your domain.