What exactly do "deposit-taking" and "deposit-taking powers" mean? How specific or general is their meaning?

Also, what are "federally authorized" deposit-taking powers?

The answer to your question may not be as obvious as it seems and can be broken down into parts.

"Deposit-taking" is part of the "business of banking." According to the National Bank Act, the business of banking includes "discounting and negotiating promissory notes, drafts, bills of exchange, and other evidences of debt; by receiving deposits; by buying and selling exchange, coin, and bullion; by loaning money on personal security; and by obtaining, issuing, and circulating notes according to the provisions of title 62 of the Revised Statutes."

Deposit taking powers are banking activities which are by and large "federally authorized" and regulated activities. The "express powers" in the National Bank Act are supplemented by "incidental" powers and banking activities, some of which are relatively "necessary" or "convenient" to the basic business of banking (including deposit-taking) and others which are not as necessary, or even "unnecessary," depending on the context.

General "deposit-taking" activities or "powers" therefore include both express and incidental deposit-taking activities.
Express activities are centered around a depository account, e.g., a checking or savings account, and its operation. Overdraft protection would be an example of an activity incidental and unecessary to basic deposit-taking.

"Deposit-taking" can also be differentiated from the area of "lending." Both are part of the "business of banking" but are separate types of activities. Deposit-taking is ruled by the uniform commercial code and regulations related to deposit accounts (e.g., Regulation CC, D, E).