For the last few years Exchange Online mailboxes have been processed by a service call Clutter, which moved the less important emails, or indeed the clutter, to a dedicated folder. This is now in the processes of being replaced by the Focused Inbox, which is client version dependant and is all based on views on the mailbox and not different folders.
The way to ensure mail is not marked as clutter, or shown in the Other view when your mailbox is processed by the Focused Inbox, is to mark the item as such, or to actively engage with the item. That is if you reply or read the emails from these recipients they do not go into Clutter/Other, but if you ignore them or delete them before they are read then this makes them candidates for future processing by the Focused Inbox or Clutter engine.
There are though times when occasional emails need to be in your Inbox and not the Other view or the Clutter folder. The best two ways to do this are as follows:
The processing engine for Clutter/Focused Inbox will not place items from your Direct Reports or management chain in the Other view/Clutter folder nor will it place any emails from yourself into the low priority location. The Direct Reports and your management chain is known to the processing engine as it is part of Active Directory. So as long as your manager (and everyone else’s manager) attribute is populated in Active Directory and synced to Azure Active Directory then this configuration can be honoured.
The other way to ensure certain messages always go to the Inbox is to have the message processed by a transport rule. Transport rules, like the management chain above are only available in Office 365 Business and not Outlook.com. The two Transport Rule placeholders below add the Clutter and Focused Inbox rules (there are two different rules, so if you added the Clutter one in the past a new one is needed for Focused Inbox). They add the rule with a arbitary placeholder, so that the rule never fires (unless you really happen to enter the demo text!). So once you add these rules change them to suit the conditions of your environment. For example if you have a “company wide communications” email sender then you could set the rule to be when that sender sends emails. The two rule placeholders you need in remote PowerShell to Exchange Online are:
1: New-TransportRule -Name "Bypass Focused Inbox" -SubjectContainsWords "This is a placeholder rule that does nothing, change this action to suit the requirements of the client" -SetHeaderName "X-MS-Exchange-Organization-BypassFocusedInbox" -SetHeaderValue "true" -Comments "<date> - <name> - Any mail that meets the conditions of this rule will go into the Inbox or Focused Inbox and not the Clutter or Other folder in Exchange Online"
2: New-TransportRule -Name "Bypass Clutter" -SubjectContainsWords "This is a placeholder rule that does nothing, change this action to suit the requirements of the client" -SetHeaderName "X-MS-Exchange-Organization-BypassClutter" -SetHeaderValue "true" -Comments "<date> - <name> - Any mail that meets the conditions of this rule will go into the Inbox or Focused Inbox and not the Other view in Exchange Online"
Change these rules to suit your requirements