I do not know when combinatorial game theory started but clearly much earlier than On Numbers and Games and Winning Ways. I have found the paper MEAN PLAY OF SUMS OF POSITIONAL GAMES by OLOF HANNER of 1959 and it references older papers.

I have a basic grammar question:

"We consider the player's start locally or in the environment."

Is it correct to use the adverb 'locally' or should it be the adjective 'local'? Is the style ok or must the sentence be rewritten?

Interesting question. I could have said, "We consider the player's local start, . . ." as start is a noun. However, start can also be considered as a verbal noun, derived from the expression, starts locally. English is funny that way. Position matters. Adjectives do not typically follow the nouns they describe, so The player's start local would also be ungrammatical in my dialect.

Predicate adjectives can also be confusing. Some people have a dialect where they can say, "I feel badly," to mean what I do when I say, "I feel bad." OTOH, we both could say, "I feel poorly," which is different from "I feel poor."

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It looks completely normal to me. We are considering the player's start. How are we doing that considering? We could consider it in a local manner or in a more global manner. We are modifying the verb "consider" and therefore use the adverb.

(I have not been following the discussion closely so if the above interpretation of the sentence is not correct then maybe the wording is indeed misleading.)