The R/3 Support Service Connection is required in nearly all situations. In this case are included memory and performance issues, any situations where a back-end error is displayed or a dump occurs. It also includes situations where a BEx Query returns unexpected results, when authorization messages are displayed, etc.
The SAP NI Connection is used for opening the queries on BEx Query Designer for analyzing their definitions. It is very helpful also to identify whether the cause of the issue is on BW side or is a matter of query definition.
WTS (Windows Terminal Server) - just in case HTTP connection is not possible. It slows down and harms the analysis process, but may be a workaround.
HANA Studio Connection - In some cases it is necessary to check some object's definitions using HANA Studio. In such cases, the HANA Studio Connection is used.
HTTP Connection - This connection is rarely used for tracing OLAP issues but it is helpful on identifying whether the issue is at BW side or at Portal's.
To update logon data, follow SAP Note 1773689 - How to add logon credentials securely to an incident - SAP ONE Launchpad (Support Portal).