Custom Button on Lightning Experience

Please refer to this: https://help.salesforce.com/articleView?id=actions_in_lex.htm&type=0

In Lightning Experience, actions display in the Global Actions menu in the header, on related lists, on list view items, and in several places on a record page. Where they appear on a record page depends on the action’s type.

[...] The page-level action menu in the record’s highlights panel (1) contains: [...] Global and object-specific quick actions, except for those related to creating tasks, creating events, and logging calls. [...]

The Activity tab (2) contains Create a Record quick actions that point to the Event and Task objects. It also contains Log A Call actions and Send Email actions.

It's a built-in Lightning page behaviour. If you have a Quick Action that creates an Activity or sends an E-mail, it winds up in that tab instead of the button area. They just don't give you any choice in the matter.

1) What you are pointing to is called as actions in the highlights section.

2) what you see there is a action not a button. You have to create a lightning action and if you want to override the actions you see in the highlight section, you will have to modify the pagelayout to include the action.

https://trailhead.salesforce.com/en/modules/lex_migration_customization/units/lex_migration_customization_actions

If you edit the pagelayout related to an user you will be able to add/remove actions from the Salesforce1 and Lightning Experience Actions section.( You will see an option to override the actions and remove the buttons that you do not want to part of the layout)

You need to create an action for the object and then add it to the page layout in the Salesforce1 & Lightning Actions section.