How far did the timing go out when you cranked it by hand? (just re-read. 4 teeth. Enough to cause damage if running but probably not by hand)
I'd say the most likely explanation is that the timing slipped, maybe you got valve to piston contact but it could well be that it just forced some oil out of a hydraulic follower and it's going to be noisy for a few minutes before it fills up.
If it was cranked over by hand a lot before starting it could be that this has just emptied the followers a little.
If you're not positive the timing was right when you put it together take the cam cover off again and double check. Make sure the idlers are torqued correctly, the timing is still OK and the tensioner marks are set correctly for a new belt. Make sure you're using the correctly numbered timing marks for cams 1,2,3 & 4. The latter two are close together on the sprockets and easy to misread. Ensure the tensioner nut is torqued then put it back together and start it again.
On a 2.6 the idler that feeds cams 3 and 4 is not adjustable so the cam timing on these can only be changed in 1 tooth increments. Sounds like it's close enough.
This time, give it a few minutes to warm up and hold it at maybe 2000 rpm for a minute or so assuming it sounds safe to run. Once the lifters have filled up it should quieten.
If not, a compression test should tell if there's any damage done.
Fingers crossed for you

Kevin