Why did Hakeem Jeffries not become Speaker when he received 212 votes while Kevin McCarthy only received 203 (see news source)? What are the exact voting rules (including the minimum number of votes, etc.) for electing the Speaker of the House of Representatives?
The requirement for the Speaker of the House to be elected by a majority vote at the start of a Congress is set not by law or the House rules, but by precedent, which is why it's quite hard to come up with a citation for the exact process. The Clerk explains as much at the start of the session: "Pursuant to law and precedent, the next order of business is the election of the Speaker of the House of Representatives for ...".
Traditionally, each party caucus nominates an individual as that party’s candidate for Speaker. Although there is no requirement that the Speaker be a Member of the House, all Speakers have been chosen from the sitting membership. A Speaker must be elected by a majority of those voting, and successive ballots are taken if no candidate receives a majority on the first ballot.
It points out in a footnote, however, that the "majority of those voting rule" is not absolute - although these uses of a plurality vote were subsequently confirmed by a majority vote:
In two instances the House chose a Speaker by a plurality of votes but confirmed the choice by majority vote. In 1849, the House had been in session 19 days without being able to elect a Speaker (no candidate having received a majority of the votes cast). Finally, after the 59th ballot, the House adopted a resolution declaring that a Speaker could be elected by a plurality. See 1 Hinds’ Precedents §221. In 1856, the House again struggled over the election of a Speaker. Ballots numbering 129 had been taken without any candidate receiving a majority of the votes cast. The House then adopted a resolution permitting the election to be decided by a plurality. See 1 Hinds’ Precedents §222. On both of these occasions, the House subsequently ratified the plurality election by a majority vote.
According to the current house rules, the Speaker of the House must be selected by a majority of votes, that is, more than half. While Jeffries had the highest vote total, he had only 212/434, which is less than half. If all 434 representatives (one has passed away since being elected) continue to vote in each round, the threshold is 218 votes. If representatives start to abstain or don't show up to vote, the threshold would be lowered.