Post History

You seem to be inquiring primarily about present-day adverbial/prepositional meanings.  (However, as your quoted resource mentions, the adverbial usage is actually older than the also mentioned conjunctive usage, and as such the adverbial usage has a closer relationship to the original "outside" both formally and semantically).

The semantic connection still exists, but it is sometimes obscured by an implicit negation.

The adverbial/prepositional "but" often follows a quantifier such as "all" or a negative "nothing" (or "nobody", etc.); and whenever it doesn't follow such a quantifier, it rather often acts as shorthand for "nothing but".

For example: 

> She is but a shadow of her past. 

...could be expanded into:

> She is nothing but a shadow of her past.

...and then to:

> She is nothing more than a shadow of her past.

...and that's synonymous with:

> She is only a shadow of her past.

The dictionary entry quoted in the OP doesn't seem to reflect the specific meaning found in phrases such as "nothing but" (where the negation is explicit, expressed separately from the remaining meaning of "but"), but that's probably because the dictionary would classify such usage as prepositional rather than adverbial.

The prepositional meaning in "nothing but" can then be compared to "nothing except" with the latter's perhaps more obvious etymological relationship between "ex-" and "out[side]".

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As a side note, the implicit negation is sometimes present and sometimes absent in "all but" contexts, depending on the part of speech represented by "but".

> The subject was all but forgotten.

Adverbial.  Positive.  "All but" means "almost", "right next to".

> We have support from all but one of the networks

Prepositional.  Negative.  "All but" means "the complement of" or "all outside of"