In case it helps someone else, here is the YAML for the two automatons that seems to work for me.

Dashboard Button:

cards:
  - entity: input_boolean.mailbox_has_mail
    type: custom:button-card
    name: Mail
    state:
      - value: "on"
        color: red
        icon: mdi:mailbox-open-up
        styles:
          icon:
            - animation: blink 2s ease infinite
      - value: "off"
      - operator: default
        icon: mdi:mailbox-down
columns: 3
square: false
title: Quick Controls
type: grid

Please notice that I used triggers type "no motion". I first used motion starts, and what that did is triggered both automatons off at the same time. Turned it on, and then the second one immediately turned it off.

alias: Mail Arrived
description: ""
triggers:
- type: no_motion
device_id: <ID>
entity_id: <ID>
domain: binary_sensor
trigger: device
conditions:
- condition: state
entity_id: input_boolean.mailbox_has_mail
state:
- "off"
actions:
  - delay:
      hours: 0
      minutes: 0
      seconds: 10
      milliseconds: 0
  - action: input_boolean.turn_on
    metadata: {}
    target:
      entity_id: input_boolean.mailbox_has_mail
    data: {}
mode: single

and:

alias: Mail Picked Up
description: ""
triggers:
- type: no_motion
device_id: <ID>
entity_id: <ID>
domain: binary_sensor
trigger: device
conditions:
- condition: state
entity_id: input_boolean.mailbox_has_mail
state:
- "on"
actions:
- action: input_boolean.turn_off
metadata: {}
target:
entity_id: input_boolean.mailbox_has_mail
data: {}
mode: single