Is there a strict geographical definition on those names from a historian's point of view? Are there differences between the geographic definition e.g. is upper mesopotamia included/excluded? I have the impression (not really sure) that while "Anatolia" includes eastern Turkey, Asia Minor doesn't extend that far. Strabo (2.5.24; 12.1.3) speaks of the region from Taurus and westwards.
According to the Online Etymology Dictionary, Anatolia is simply "the ancient name of Asia Minor".
However, here is some helpful background on the different origins of the two terms. In theory this could be seen as implying a different definition for Asia Minor, but in practice the terms are synonymous.
Asia Minor is a geographic region in the south-western part of Asia comprising most of what is present-day Turkey... It was called, by the Greeks, “Anatolia” (literally, 'place of the rising sun’, for those lands to the east of Greece). The name 'Asia Minor’ (from the Greek `Mikra Asia' - Little Asia) was first coined by the Christian historian Orosius (c. 375-418 CE) in his work Seven Books of History Against the Pagans in 400 CE to differentiate the main of Asia from that region which had been evangelized by the Apostle Paul (which included sites known from Paul’s Epistles in the Bible such as Ephesus and Galicia).
As for a precise definition, here's one from Encyclopedia.com:
Anatolia, also known as Asia Minor, is a large, mountainous peninsula of approximately 755,000 square kilometers (291,500 square miles) that extends from the Caucasus and Zagros mountains in the east and is bordered by the Black Sea on the north, the Aegean Sea on the west, and the Mediterranean Sea on the south. It comprises more than 95 percent of Turkey's total land area.
Wikipedia also treats the two terms as synonymous, but gives a slightly different definition:
The article goes on to explain that indicating the precise boundary of Anatolia is politically fraught in modern times due to tensions between Turkey and Armenia.