I'm more familiar with medieval maps then those from later on so my answer will be a little limited, unfortunately. First of all, different civilizations used different directions as 'up' on their maps. West was actually quite rare so that Phoenician example is really interesting. Christian Europe generally used East as 'up' since Jerusalem was to the East. Islam had a tradition of using South as 'up' since for much of the early Islamic community (pre conquest of North Africa) Mecca was to the South and the tradition just stuck.

China used North as 'up' since the Emperor was to the North of where the majority of the Chinese population lived. I'm not sure what the earliest Chinese map to use North as 'up' is, but it's pretty old. The Kagnido Map is actually a Korean map but it is based in a Chinese cartographic tradition and is the earliest complete map I'm aware of that shows North at the top.

For Western Europe the Mappa Mundi, for an example see The Hereford Mappa Mundi, of the middle ages slowly began to be replaced by the type of maps we would be familiar with in the sixteenth century. These newer maps, which the Waldseemuller Map is a good example of, did not completely replace Mappa Mundi until the seventeenth century but they do become increasingly common as the sixteenth century progressed. Exactly why they changed for the North as 'top' orientation I'm not entirely sure. The obvious catalyst is Columbus voyage to the west but even more broadly the increasing amount of exploration happening in the fourteenth century with expeditions like de Gama's in Africa meant that sailors increasingly needed maps to know where they were going. While Mercator's Projection which actually made for practical navigation by map wouldn't be made until 1569 there was an increasing trend to replace the very religious and impractical Mappa Mundi with a more practical and 'accurate' series of maps of the world. Why these latter maps chose for North to be 'up' I'm honestly not sure. If I were to speculate I would guess that it had something to do with the Ptolemaic projections they were using which started at a single point (one of the poles) and projected its grid outward from there in an attempt to capture the spherical globe on a flat surface. While in theory you can totally make one of these with East as 'up' (you just put your starting point on the right) it does seem like it would easier to just keep the North pole at the top as you did this.

Jerry Brotton's A History of the World in 12 Maps is definitely worth a read if you're interested in the history of maps. It's title makes it sound like it's a World History that uses maps as an opening for further discussion but it's actually a history of cartography from prehistory to the modern day.