For just the reasons that you describe, one can't put total trust in the numbers on the 1098-T to use the reported numbers directly for the yearly tax return. Schools are not consistent, and the tuition numbers that schools report are ambiguous with regard to billing and payments for spring semesters and winter quarters.
As you correctly note, IRS Pub 970 specifically states that education payments count in the year that you pay them. For the current spring semester or winter quarter, if you wrote and mailed your check in December 2018, then that counts for 2018. If you waited to write and mail your check until January, then it would count for 2019. There has never been a good way to capture that change-of-year issue using the 1098-T. So just keep your own financial records and copies of your checks or other payment methods, and use your own accurate information. You will need your own financial records for other qualified items, such as books for the American Opportunities Credit since those are not on the 1098-T. In addition, obtain and save copies of the school's billing and payment ledger in order to have the financial records that can substantiate your claims and verify your payments and the expense items.
Put the true numbers into your tax return based on the true dates of the payments. Don't worry that the 1098-T isn't a 1-for-1 match. I hardly ever see them match on any consistent basis. If you ever get questioned, then use your financial records and the school financial ledger to provide the proof for your return.
The good thing about the 1098-T is that it does indicate that the student was enrolled at the institution, and it will indicate whether or not the student is attending as at least a half-time student--that is important information. Taken together over a series of years, the 1098-T forms can provide the IRS with a good picture of tuition expenses to ensure that your credit claims are reasonable.