We don't even know for sure that the Atraxi got the images from some kind of databank, rather than that Doctor's memories. But, even assuming they did, it's not too hard to put the Doctors in order.
In general, modern-day TV episodes (and even novels, etc.) take place the year they're broadcast (or published). And none of the exceptions pose a difficult problem.
- The 1st Doctor rarely visited modern Earth, but he spent 6+ months in 1963 before the series started (and even got a library card), so he's easy to find.
- The 2nd Doctor only had one major visit to modern Earth, "The Invasion", but it was clearly after 1963.
- The 3rd Doctor stories are hard to place (see tardis:UNIT dating controversy). But the UNIT stories are obviously after "The Invasion".
- The 4th Doctor must be after the 3rd, because of Sarah Jane. With a far-future-enough dating for the UNIT stories, you have to conclude that the 4th Doctor picked up Tegan from earlier than he dropped off Sarah Jane, but that doesn't affect the ordering.
- The 5th Doctor's modern stories were all in the present. And he must be after the 4th, because of Tegan.
- The 6th Doctor's modern stories were all in the present.
- The 7th Doctor's modern stories were all in the present, except for the TV movie.
- The 8th Doctor's only story was 3 years in the future, but since that doesn't overlap anyone else, that's not a problem.
- The 9th Doctor's stories were 1 year in the future, but they don't overlap anyone else.
- The 10th Doctor's stories were 1 year in the future, but they don't overlap anyone else.
- The 11th Doctor's stories are in the present (except the series 7 Christmas special, where he was 2 years in the future). They may slightly overlap the end of the 10th Doctor, but it's pretty easy to tell that 2007-late 2010 comes before mid 2010-2013.
If this were in a book of logic puzzles, it would be in the beginner's section.
Uh... hang on. He can time travel so that doesn't prove anything. I know you said "In general, modern-day TV episodes (and even novels, etc.) take place the year they're broadcast (or published). And none of the exceptions pose a difficult problem", which is very true, but unless the Atraxi have fourth-wall breaking powers HOW WOULD THEY KNOW THAT?