Three verbs make the sequence clear.
The nomination identifies a person for parliament to vote on. Election is the Bundestag's parliamentary decision. Appointment by the Federal President is a separate formal act.
Although election campaigns discuss candidates for Chancellor, the Bundestag ballot does not contain a separate direct election for that office. Eligible voters determine parliament's composition.
The other three answers
None of these alternatives replaces the Bundestag election.
the people
The people elect the Bundestag but do not directly elect the Chancellor. A popular candidate has not thereby been legally elected.
the Federal Assembly
The Federal Convention elects the Federal President. It does not fill the chancellorship.
the federal government
The federal government cannot elect its own Chancellor. That electoral power belongs to parliament.
A failed nomination does not change who conducts the election.
Article 63 provides further stages if no candidate initially obtains the required majority. Those ballots also take place in the Bundestag; rules for nomination and appointment vary by stage.
Political agreements and parliamentary decisions are connected but are not legally interchangeable. A party resolution is not a Bundestag election of a Chancellor.
Article 63 · Basic Law ↗Nomination – election – appointment: election belongs to the Bundestag.
The verb in the question determines which responsibility is being asked about.
Three words for this question
- vorschlagento nominate
- To name a person for a subsequent election.
- wählento elect
- To select a person for an office by voting.
- ernennento appoint
- To confer an office through a formal legal act.