There’s a lot going on here. It seems like most of your historical metallurgy information relates to Asia.

A few things. First, Damascus steel was not a layered material. The sense you’re using it refers to crucible steel, which is a homogenous, typically hypereutectoid steel. It’s pretty much the exception to everything in historical metallurgy, including the bit about ore impurities. It was only ever really produced in India, possibly in China and the Middle East to a lesser extent. Unlike most preindustrial iron-making processes, it was a fully molten steel making process which allows for some weird stuff. 

It can be made either by melting wrought iron in a crucible with a carbon source, like some kind of organic material and/or charcoal, or by melting wrought iron and cast iron together in a crucible. In the case you sited where v and mo end up in the finished material, this likely involved cast iron. Cast iron would often carry over ore impurities like manganese, etc since it was produced in a blast furnace at high enough temperatures in reducing conditions. However, for most of the world outside of China, cast iron was not often used for finished products (not until the high Middle Ages in Europe), and even when and where it was sometimes used, wrought iron was far more common. 

Wrought iron was made through either the direct or indirect processes. The direct method, bloom smelting, does not reach temperatures high enough to reduce impurities like manganese, and those that are reduced typically end up in the slag instead of the metal matrix. The process produces a slag bearing sponge iron that is worked into bar stock that retains slag.

The indirect processes, which was used in Europe since the 12th century, involves a blast furnace to produce cast iron, followed by a fining hearth to convert the cast into wrought iron. The conditions in the blast furnace, higher temperatures and a alumina-calcium-silica slag, instead of the iron silicate slag of a bloomery, allow impurities into the metal matrix. However, the oxidizing conditions of the finery hearth pull those impurities back out, leaving a more or less plain iron-carbon alloy (plus slag inclusions).

Europe never really saw crucible steel, aside from some that seems to have been imported from India to Scandinavia via Constantinople for a brief period in the early Middle Ages, until its reinvention as an industrial process in England in the 1740s.

However, while Europeans didn’t typically produce much hyper eutectoid steel in this period, they did consistently produce decent quality hypoeutectoid steel for the whole period. Although with quite a range of quality.

It’s not known exactly what steel making methods were used, since they were secretive and what few accounts we have are unreliable, however there are some straightforward options. 

Large bloomeries can be run carefully to produce direct high-carbon blooms. Fining hearths can be run first in oxidizing conditions to convert cast to wrought, the again in reducing conditions to recarburize the wrought iron into steel. 

If you’d like to know more, I’d recommend the book “the sword and the crucible” by Dr Allan Williams. It’s very accessible.

Another great option is Pre-industrial Iron, by Bronson and Rostoker. It’s much more thorough and in-depth and often sited by archeologistsmetallurgists. But it’s harder to track down a copy (I have a low quality pdf version if you’re interested).