There is a pull request to add this feature though:
That pull request sets the DHCP identifier. That’s not what OP was requesting.
OP tries to send requests to he’s Arduino using mDNS host names. You can use this library to let your Arduino answer mDNS requests and reserver a local host name for it. Keep in mind that this works only for clients that implement mDNS (Android for example doesn’t provide that feature by default).
How one can set DHCP name using Ethernet API? I cannot find setHostname() or similar.
No, the API doesn't allow that. The mentioned pull request would add such a feature to the Ethernet library but it wan't merged yet as far as I know.
The pull requests you mentioned are even older and were probably replaced by the one above.
I can understand that arduino.cc is cautious with the pull requests as it might pose a lot of problems in using the library for a functionality very seldom needed (other than for narcissistic reasons).