Please see Publication 970, Chapter 2, Page 14, left column. The rule is buried in the AOC section, but it applies to LLC too. It states, "
Qualified education expenses paid in 2018 for an academic period that begins in the first 3 months of 2019 can be used in figuring an education credit for 2018 only. See Academic period, earlier. For example, if you pay $2,000 in December 2018 for qualified tuition for the 2019 winter quarter that begins in January 2019, you can use that $2,000 in figuring an education credit for 2018 only (if you meet all the other requirements)."
Then look at the big CAUTION SYMBOL that more clearly defines this issue that says, "You can't use any amount you paid in 2017 or 2019 to figure the qualified education expenses you use to figure your 2018 education credit(s)."
You can also look at Chapter 3 for the Lifetime Learning Credit on Page 24, right column where it says, "Generally, the credit is allowed for qualified education expenses paid in 2018 for an academic period beginning in 2018 or in the first 3 months of 2019. For example, if you paid $1,500 in December 2018 for qualified tuition for the spring 2019 semester beginning in January 2019, you may be able to use that $1,500 in figuring your 2018 credit."
So to translate, if paid in 2018, then use it in 2018. If paid in 2018 for a winter/spring session, then the payment is deemed to have been paid for a qualified education expense. Even though the school session might be 2019 it is deemed to be OK for a 2018 expense. The text and caution symbol and the examples make this fact abundantly clear.
The IRS acknowledges this problem in Pub 970 on Page 3 where it states, "However, the amount on Form 1098-T might be different from the amount you actually paid and are deemed to have paid."
Therefore, when claiming payments for tuition credits, one should use the truth and should not claim the credit using incorrect information. The IRS rule is to use the date of the payment. It is OK to pay in December and claim in that year because the IRS deems that to be OK.
I generally see people taking the AOC, this is usually not a problem because the credit tops out after spending $4000 on tuition. For most schools, that is only one of the two semesters. So when the 1098-T gives $4000 or more, and if one really did pay the $4000 or more for tuition and books in the tax year, then we can just enter the $4000 number for AOC tuition, and we move on.
And while books are not qualified educational expensed for LLC, BOOKS ARE QUALIFIED EDUCATIONAL EXPENSES FOR AOC. Housing is not qualified for either LLC or AOC, but housing is qualified (with some limitations) for 529 spending.