Well there's a matter of timing; I gather the story about carrots was particularly connected with "Cat's Eyes" Cunningham who was flying a Beaufighter with early Airborne Intercept radar in November 1940, whereas the first German and Japanese airborne units went into service in 1942.

In more general terms, though, it's not so much the existence of radar that was secret (there was a lot of research around the world in the 1930s) as the precise nature and capabilities of particular systems, what we now call Electronic Warfare. I doubt the carrot story was a particularly serious effort, but anything that might cause a bit of confusion or throw the enemy off the scent is worth a try.

As you mention, there were several different types of system in use. Both sides used guidance systems, especially for night bombing, such as the German Knickebein and X-Gerät, and British GEE and Oboe; in turn each side would try and discover, jam or misdirect the enemy systems, the "Battle of the Beams", well detailed in Most Secret War by R V Jones. Airborne radars could be used for detecting other aircraft (Airborne Intercept, AI, in the RAF) or ships (Air to Surface Vessel, ASV), or for mapping ground contours for navigation/bombing (H2S); in turn these radars could be detected or jammed by the enemy (the RAF fitted a rear-facing radar known as "Monica" to bombers, intended to warn of enemy night fighters, but once the Germans were aware of it they developed a receiver for their night fighters that could home in on the signals). In all cases the technical specifications were crucial, wavelengths, power and such; electronic equipment from aircraft that were shot down or captured was greatly prized (the Junkers Ju 88 R-1 at the RAF museum, fitted with Lichtenstein AI radar, was flown to Britain in 1943 by a defector).

One of the reasons that the Germans were slower to develop airborne radar was, ironically, that their land-based systems were more effective. British radar stations (Chain Home) were famously crucial during the Battle of Britain as part of the RAF command and control system, but technologically comparatively crude so there was a greater impetus to develop the AI sets. One of the most important Allied developments was the cavity magnetron, a device that allowed (relatively) small and lightweight sets operating at shorter wavelengths, thus capable of detecting smaller objects such as a submarine periscope as opposed to just a ship. Until very late in the war German airborne radars, such as the Lichtenstein unit on the aforementioned Ju 88, needed the distinctive large nose-mounted aerials that degraded the performance of the aircraft, whereas by 1943 the RAF A.I. Mk VIII centimetric radar fitted neatly into a radome in the nose of the Beaufighter and Mosquito.

Other sources:

Technical and Military Imperatives: A Radar History of World War 2, Louis Brown

Array and Phased Array Antenna Basics, Hubregt J. Visser

I would add that there was an element of strategic opportunism in the Japanese not knowing the technical capabilities of British radar. The Americans had broken the Japanese coding system, so they knew where and when the Japanese would be in battles. Add in radar and they could pinpoint even more accurately where Japanese fighters were. If the Japanese were aware of such abilities they would have adjusted their battle strategies and planning to compensate for radar and, obviously, would have changed their coding system.

Speaking for the Japanese "radar," I can attest that it looked and functioned like crap compared to British and American models.

For instance, the US were able to deploy fire-control radar well in advance of anything Japan was able to deploy. Essentially, using radar, USN ships were able to accurately fire shells at enemy surface vessels by having the appropriate trajectories calculated using the radar's data. On the other hand, the Japanese were mostly reliant on optical rangefinders-in effect, iron sights and binoculars-to engage the enemy in surface combat. This was most proved effective during the night skirmishes throughout the Solomons Campaign, where despite superior Japanese crew quality and torpedoes, the US radar-guided guns were able to accurately wreak havoc on the Japanese ships.

Another major impact was the US development of proximity fused shells, which utilized radar. The shells were able to detect their approach near a plane and detnoate, thus increasing their effectiveness against the plane many times over. Whereas Japanese anti-air guns had to actually hit the plane for the shells to detonate, American ones only needed to be near the plane to do so, with the ensuing shrapnel and explosion dealing damage to the lightly-armored Japanese planes. Had the Japanese acquired any of this technology, while their industrial weakness would have made it fairly unlikely that it would ever be deployed, it could potentially have increased Allied losses.