Ummm. the operative words are "designed and built as Fleet Carriers". HMS Ark Royal was launched 13 April 1937, and commissioned 16 Nov 1938.
Pre-dating Ark Royal into fleet service as designed and built for purpose fleet carriers were USS Ranger, USS Yorktown, and USS Enterprise. Ranger was launched 25 February 1933, and commissioned 4 June 1934. Yorktown was launched on 4 April 1936, and commissioned on 30 September 1937. Enterprise was 3 October 1936, and commissioned 12 May 1938. I suppose one could present a convincing argument that Ranger, though designated a fleet carrier, was, in the long view based strictly on tonnage, somewhat smaller than a fleet carrier, but larger than a light carrier . . . somehere in between. Damn those treaty restrictions!!!
HMS Hermes, as Andrew noted was a designed and built for purpose aircraft carrier, but would tend to be included in the Light Carrier category on the basis of size, tonnage, and, most importantly, complement.
The first Japanese carrier was Japanese IJMS Hosho, usually noted as the first design and built for purpose aircraft carrier, Hosho, in fact, started life as a tanker, laid down 16 Dec 1919, redesignated as a carrier 13 October 1921, launched 13 Nov 1921, and completed 27 Dec 1922. Hosho, like USS Langley of about the same vintage (renamed and converted from the collier USS Jupiter, starting around 11 April 1920 and recommissloned 20 March 1922), was essentially an experimental carrier. Hermes could also tend to fall into the experimental category if you want to stretch the definition a bit. Also Hermes has the dubious distiction of being the first aircraft carrier sunk by aircraft from an enemy carrier (9 April 1942). Langley, sunk on 27 February 1942, had already been converted to a seaplane tender and was serving as an aircraft transport when sunk by land based aircraft. Of these three experimental carriers, then, only Hosho survived to the end of WW2.
Regards,
Rich