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UPDATE 7-U.S. crude, Brent hit 3-week highs on Cushing, gasoline draw

* U.S. crude stocks fall in Cushing but rise overall

* Russia backs off demands for gas payment from Ukraine

* Libya's western El Feel oilfield resumes output -NOC

* Protesters still stop restart of Libya's Sharara oilfield

(Rewrites throughout, updates with settlement prices)

NEW YORK, May 14 (Reuters) - Global oil prices rose on Wednesday as a draw on U.S. crude stocks at the Cushing, Oklahoma, contract delivery point and gasoline stocks pushed Brent and U.S. crude to three week highs.

U.S. crude stocks rose overall last week but inventories at the Cushing, Oklahoma, delivery point fell by 592,000 barrels, data from the U.S. Energy Information Administration showed on Wednesday.

Gasoline inventories fell by 772,000 barrels along with distillates, which were down 1.1 million barrels, as refineries cut output or went down for maintenance before the official start of the summer driving season just two weeks away.

The Ukraine crisis eased slightly as Russia backed off its threat to cut Kiev's natural gas supply if the country did not pay in advance for June. It instead asked that Kiev pay just part of the "debt" Moscow it owes.

The conciliatory tone did little to reassure European Union leaders, however, who called the demand "unjustified." Half of Europe's imports of Russian natural gas go through Ukraine.

Brent crude for June delivery extended gains for a second day to settle 95 cents higher at $110.19 a barrel, its highest settlement since April 24. The contract expires Thursday, a factor that may have exaggerated price gains because there is a lower volume of trades.

The July contract for Brent, which will become the front-month contract on Friday, rose 77 cents to settle at $109.31 a barrel.

U.S. oil settled 67 cents up at $102.37 a barrel, its highest settlement since April 21.

New York gasoline prices rose nearly 4 cents to $2.9693 a gallon, its highest settlement this month.

"We saw strong gasoline demand data in the report this morning, which combined nicely with the reduced oil stocks at Cushing to encourage futures prices higher," said Richard Hastings, a macro strategist at Global Hunter Securities.

Traders have also been eyeing Libya, where state officials and rebel leaders have gone back and forth on agreements to reopen several oilfields.

The El Feel oilfield, also known as Elephant, resumed production, Libya's state oil company said on Wednesday, after protesters agreed to end their blockade of the pipelines connecting that field to the port of Mellitah.

But the larger 340,000-barrel-per-day El Sharara oilfield was still closed as its pipeline to the Zawiya port remained blocked.

China remained a background factor that dampened demand after data released Tuesday showed softness in the world's second-largest economy in April. Oil demand has been slowing there since mid-2013, the International Energy Agency said in a report.

(Additional reporting by Julia Payne in London and Keith Wallis in Singapore; Editing by William Hardy, James Dalgleish and Chizu Nomiyama)