Yes, it depends on the width of the rim. If the rim is 10 inches wide, a 12.4x38 will fit nicely. If the rims are 11 inches wide, then go with the 13.6x38.
The 12.4 more or less replaced the old 11.2, which replaced the original 10.0.
Yes, you can squeeze a 13.6 onto the 10 inch rim, but it will have a really odd look to it. You can also put a 12.4 on an 11 inch rim, but again, it will look really odd.