Conditions in which you can handle the boat depend on the style of boat and the paddler
The range of what defines a "covered cockpit kayak" includes play boats from under 2 meters to over 5 meter touring and racing boats. A considerably greater variety than "open canoe" covers.
Paddlers range from total novices to the highly experienced or internationally competitive.
At one end of these scales you're going to have serious difficulty travelling upwind at all under these conditions, at the other end of the scale you're unlikely to really notice until conditions become significantly more adverse.
Advice for each combination of boat and paddler varies. One thing you will notice is that kayaks tend to turn nose into the waves, making upwind an easier option than you might expect.
The easiest way off for a short boat will always be downwind, these boats are designed to surf. They'll catch a wave and you can ride it to shore. They really struggle to climb waves and you won't make much progress if you turn into the wind and waves.
Crosswind requires some skill at handling your boat in waves, but if you drop the edge towards the incoming waves and let it turn up the wave and ride down the back it's not too bad. The chances are anyone needing this advice will be in a stable enough boat that there's not a massive problem but should generally be avoided. Taking waves straight side on should also be avoided, however as already mentioned, your boat will naturally turn up into the waves as long as you keep it upright.
Upwind could be your better bet in longer touring boats, racing boats or sea kayaks. The boat will naturally turn this way and it won't cause any difficulties with handling. However these boats can also surf, so a good paddler should be able to take a quick ride out by going downwind. Anything above the 4 meter mark is unlikely to be slowed by small white horses, smaller touring boats will get wetter, but the point of the covered cockpit is to allow you to stay out in considerably more adverse conditions than open boats can handle.
In summary: A competent kayaker in a suitable boat doesn't actually need to head off the lake at the point where white horses are starting to form, however beginners should take an exit that suits their abilities and their boat.
The plural of anecdote isn't data: As a general rule once the white horses start appearing I treat it like a resistance session and dig in. I'm usually in a polo boat which isn't designed to handle at all in big water, however I'm also not your average recreational paddler. I advise beginners to stay away from areas where the waves tend to build when the wind picks up.
Storm in a teacup
Don't try this at home!
I was thinking of this question as I was out on the lake on Sunday morning, the image is the output of the weather station on our base.
Since we don't get conditions like this very often, I took the opportunity to test both long and medium boats. (I don't fit well in the small boats, and in these conditions I wasn't going to take anything unfamiliar.)
The sea kayaks (P&H Delphin) were great both up and downwind, surfed nicely with the wind behind us (skeg down), coming off the surf at speed onto the crosswind leg with limited space was less fun, but they were also no problem at all upwind.
Polo boat (EXO XP3) 3metres low volume. This is not a boat I'd normally recommend anyone take under such conditions, it's fundamentally a flat water boat. Downwind it was a handful in the surf, constantly trying to broach (no skeg of course) and preferring to be half submerged if at all possible. In big water you want to keep your nose up and on this boat any lift under the tail the nose goes down (feature not a bug). Coming onto the crosswind leg it started behaving itself. Upwind it tends to sit level and let the waves roll over the top. That's not actually a problem and it paddles nicely even half underwater.
This is as much an example of the difference between closed kayaks and others such as sit on tops, open canoes and paddleboards. Only a closed kayak could be used even under our controlled conditions in the storm. Any other type would have washed up battered on the lee shore in fairly short order.