For some reason I get answers that are very close but not equal to the ones in example:

14.495347194
0.000000000
56.489368526

Maybe I misunderstood something in the model?

There's no any day of year where you can get such answers with latitude = 56.
Only if the time differs about 15-20 seconds from the time of original test data.
So it seems you choose slightly different Vasya's location because of wrong time calculation.
Or maybe the proximity to the correct answers is just a coincidence, and your Earth-moving model is totally wrong.
What answer do you get for the test
56 24
21 jun 0
21 jun 1
...
21 jun 23 ?

At least they are symmetric and reach maximum at 12 hours...
Could you please post the correct coordinates for 0 and 12 hours and I'll go from there.

0.000000000
0.000000000
0.000000000
0.000000000
4.214527110
11.374704540
19.267892795
27.563494585
35.903413441
43.838560055
50.725478497
55.623901463
57.439281000
55.623901463
50.725478497
43.838560055
35.903413441
27.563494585
19.267892795
11.374704540
4.214527110
0.000000000
0.000000000
0.000000000

Edited by author 11.05.2015 16:35

Edited by author 11.05.2015 16:47

Wow, they're again very close to my answers. Here they are:
0
0
0
0
4.2133420379
11.3735281948
19.2668088249
27.5625655432
35.9026841855
43.8380585719
50.7252091429
55.6238233024
57.4392810000 (totally correct :))
55.6238233024
50.7252091429
43.8380585719
35.9026841855
27.5625655432
19.2668088249
11.3735281948
4.2133420379
0
0
0

I have no idea what exactly it could be, because any misunderstanding of the model would cause much greater error in the answers. Check the constants you use. Maybe one of them has little inaccuracy or doesn't have enough precision.

Found my mistakes. I handled rotation around Earth's axis totally wrong.

First, its period is not 24 hours but a little shorter.
Second, the fact about closest meridian at midday doesn't imply that Vasya's location is the closest point to the Sun at his latitude.