An enterprise geodatabase is the appropriate choice as it meets all the specified requirements. It is stored in a relational database management system (RDBMS) such as SQL Server, Oracle, or PostgreSQL, which is installed on the organization's local servers. This architecture is specifically designed to support multi-user editing via web services (e.g., feature services) and provides the full capabilities of the underlying RDBMS. This includes creating custom database indexes on specific business fields to optimize query performance, which is a key requirement for the organization.

B. File geodatabase: This is not ideal for multi-user editing environments. It lacks the robust transactional and performance-tuning capabilities, such as creating custom database indexes, that an RDBMS-based enterprise geodatabase provides.

C. Hosted relational database: The term "hosted" typically implies that the data is managed by a third party or in a cloud environment (like ArcGIS Online), which contradicts the requirement for data to be stored locally on the organization's servers.

1. Esri, ArcGIS Pro Documentation, "What is an enterprise geodatabase?": "An enterprise geodatabase is a geodatabase in a relational database management system (Oracle, Microsoft SQL Server, PostgreSQL, or IBM Db2). Enterprise geodatabases can be used by many users at once and support versioned editing and replication." This supports the multi-user editing and local RDBMS storage requirements.

2. Esri, ArcGIS Pro Documentation, "Types of geodatabases": This document compares geodatabase types, stating, "Enterprise geodatabases... are primarily used in multiuser or enterprise settings where the geodatabase is centrally located and accessed by many users." This contrasts with file geodatabases, which are recommended for single users or small workgroups.

3. Esri, ArcGIS Pro Documentation, "Attribute indexes": "In an enterprise geodatabase, you can create additional database indexes on fields to improve query performance. You can create an attribute index on a single column or on multiple columns." This directly addresses the requirement to index specific business fields.

4. Esri, ArcGIS Enterprise Documentation, "Data and publishing in ArcGIS Enterprise": In the section "Data sources," it clarifies that to publish from your own on-premises RDBMS, you must register it as a data store. This RDBMS contains the enterprise geodatabase, fulfilling the local storage and web service editing requirements.