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This other question asks whether the House requires a vote to begin an impeachment inquiry, but several of the answers (including the accepted one) point out that the rules have changed since Clinton's impeachment, and thus a vote by the whole House is no longer required. However, I don't see a what or why for those rule changes.

Aside from the expiry of the position of Independent Counsel, what House rule(s) have changed to permit an inquiry-without-vote? When were they changed, and what reasons (if any) were cited for doing so at the time?

As the answers in Does the US require a House vote to begin an impeachment inquiry? point out, House rules never required a vote to start an impeachment inquiry. In both previous impeachments, however, the House did pass an inquiry resolution in order to give themselves additional subpoena power in order to better carry out the inquiry.

This is not necessary now, however, as House rules already give the majority unilateral subpoena power. CNN states that this actually dates from rule changes made by the previous Republican controlled House:

In addition, Pelosi doesn't need the House vote authorizing an inquiry because her caucus already has extra legal authority compared to past inquiries.

During the Clinton and Nixon impeachment inquiries, the House passed their inquiry resolutions so they could gain tools like more subpoena power and depositions, and included in those resolutions were nods to bipartisanship that gave the minority party subpoena power, too.

But the House rules have changed since the last impeachment of a president more than two decades ago. In this Congress, the House majority already has unilateral subpoena power, a rule change that was made when Republicans last controlled the House, so Democrats don't need to pass any resolution to grant those powers.

https://www.cnn.com/2019/10/08/politics/nancy-pelosi-letter-impeachment/index.html

An article on Lawfare provides more detail on the specific rule changes involved:

What Powers Does a Formal Impeachment Inquiry Give the House?

The impeachment proceedings against both Presidents Nixon and Clinton began with a vote by the full House of Representatives ... In both cases, the resolution granted several specific powers to the committee for it to use in the course of completing the investigation with which it was charged by the full House.

...

until recent years, unilateral subpoena power was relatively rare for House committee chairs. But between the 113th and 114th Congresses, the number of chairs given this power by their committees doubled—and the judiciary committee was among them. The judiciary committee chair retains this authority in the current Congress; its rules stipulate that “a subpoena may be authorized and issued by the Chairman … following consultation with the Ranking Minority Member.”

...

Under practices in place in 1974 and 1998, deposition power for committee staff was periodically authorized by the full House for the purpose of specific investigations. ... Since 1998, however, the rules of the House governing staff depositions have evolved to give committees access to the tool more regularly. In 2007, the House Committee on Oversight and Government Reform was given the ability to set its own rules “authorizing and regulating the taking of depositions by a member or counsel of the committee.”

...

In 2017, the rule permitting staff depositions was extended to cover almost all standing committees, and the member attendance requirement was modified such that it did not apply if the committee authorized the staff deposition to take place when the House was not in session. ... So the judiciary committee already has the power to conduct staff depositions and does not need a special grant of authority to do so.

A tiny bit more detail from ABC:

The GOP minority does not have subpoena power, much to their chagrin, although Republicans changed the playbook in 2015 when they rewrote rules delegating subpoena power to individual chairmen without full approval from the House of Representatives.

Democrats reaffirmed that process earlier this year, adopting impeachment rules that includ[e] granting their chairmen the right to issue subpoenas unilaterally.

So I think that's the change that CNN is referring to. I haven't yet found the exact text of the decision, but we know its substance and year now.

And the exact rule in which this change is incorporated is

Rule XI, clause 2(m)(3) also allows committees to adopt rules to delegate the authorization and
issuance of subpoenas to a committee’s chair “under such rules and under such limitations as the
committee may prescribe.” This same subparagraph requires subpoenas to be signed by the chair
or by a member who has been designated by the committee.

[W]hat House rule(s) have changed to permit an inquiry-without-vote?

Rules X and XI.

Must the House Vote to Authorize an Impeachment Inquiry?, October 9, 2019:

The House has changed its internal procedures dramatically over time. At one point, the House did not rely on standing committees but instead created select committees to handle many legislative tasks. Through much of its history, the House has limited the investigatory powers of its standing committees and required that those committees go to the floor to receive special authorization to issue subpoenas or spend substantial resources on staff.

Because there is no longer a need to go to the floor, there is no need for a vote to conduct investigations (inquiries). Such votes, prior to the change, would have gone to the Committee on Rules (see below) to establish select or special committees.

When were they changed[?]

October 8, 1974, to be effective January 3, 1975. [Later changes were not researched.]

[W]hat reasons (if any) were cited for doing so at the time?

To reform the structure, jurisdiction, and procedures of the committees of the House of Representatives. The Congressional Record (see below) was not instructive.

Basic information

H.Res.988 — 93rd Congress (1973-1974)

Sponsor: Rep. Bolling, Richard [D-MO-5]

03/19/1974: Introduced in House

10/08/1974: Passed/agreed to in House: Measure passed House, amended, roll call #589 (359-7)

Official Title as Introduced: Resolution to reform the structure, jurisdiction, and procedures of the committees of the House of Representatives by amending rules X and XI of the Rules of the House of Representatives.

Summary: ... Revises Rule XI of the Rules of the House of Representatives with regard to the rules of procedure for committees, including those rules related to regular meeting days, committee records, proxies, investigative hearings, and reporting of bills and resolutions. ... [Only the most relevant item shown.]

Committee on Rules

Jurisdictional Commentary From the House Rules and Jefferson Manual The Committee on Rules

The jurisdiction of this Committee is primarily over propositions to make or change the rules ..., for the creation of committees ..., and directing them to make investigations ... . Effective January 3, 1975, however, the authority for all committees to conduct investigations and studies was made a part of the standing rules (clause 1(b) of rule XI), as was the authority for all committees to sit and act whether the House is in session or has adjourned, and authority to issue subpoenas (clause 2(m) of rule XI) (H. Res. 988, 93d Cong., Oct. 8, 1974, p. 34470).

Congressional Record

Congressional Record -- House October 8, 1974, pp 34407-34470. [This includes an address by President Ford to a joint session of Congress. pp 34421-34424]

A cursory reading of the text shows that, for the most part, the representatives wanted to submit amendments to the amendment to either enhance or protect their committees. In other words, there is little in the discussion to reveal a purpose for the changes, other than the Official Title given above.