I have personally confirmed this with a keystone + Swift dev environment, while reviewing patch https://review.openstack.org/#/c/454386:

firstly the default values are defined:
<snip from Colleen Murphy's comment in above patch>

The name of the default domain is defined here:
http://git.openstack.org/cgit/openstack/keystone/tree/keystone/resource/core.py?id=69d03670d1ddcf428063d81a0f25b6196650876e#n841

The default ID of the default domain is defined here:
http://git.openstack.org/cgit/openstack/keystone/tree/keystone/conf/identity.py?id=69d03670d1ddcf428063d81a0f25b6196650876e#n21
</snip>

So params ending in _id are default and the -name are Default.. which I find annoying. But there you go. Our sample doc uses _id = default which "should" be correct.

What do others use, well according to https://docs.openstack.org/developer/keystonemiddleware/api/keystonemiddleware.auth_token.html
it gives a sample, apparently from nova that:

  [keystone_authtoken]
  auth_plugin = password
  auth_url = http://keystone:35357/
  username = nova
  user_domain_id = default
  password = whyarewestillusingpasswords
  project_name = service
  project_domain_id = default

Which using using "default" against the *_id versions of the configuration.

Now if I pop into my test keystone server, and take a look:

  (openstack) domain list
  +---------+---------+---------+--------------------+
  | ID      | Name    | Enabled | Description        |
  +---------+---------+---------+--------------------+
  | default | Default | True    | The default domain |
  +---------+---------+---------+--------------------+
  (openstack) domain show default
  +-------------+--------------------+
  | Field       | Value              |
  +-------------+--------------------+
  | description | The default domain |
  | enabled     | True               |
  | id          | default            |
  | name        | Default            |
  +-------------+--------------------+

Now lets do some keystone config tests in Swift.
First using:

  [filter:authtoken]
  paste.filter_factory = keystonemiddleware.auth_token:filter_factory
  auth_url = http://localhost:35357
  auth_uri = http://localhost:5000
  auth_plugin = password
  project_name = service
  username = swift
  password = Sekr3tPass
  delay_auth_decision = True
  user_domain_id = default
  project_domain_id = default

It works:

  $ swift stat
                 Account: KEY_fce1546e0675483bb3862a54a2ed68f6
              Containers: 0
                 Objects: 0
                   Bytes: 0
         X-Put-Timestamp: 1491904442.67731
             X-Timestamp: 1491904442.67731
              X-Trans-Id: tx0814c8d8f3694f47a13d9-0058eca7ba
            Content-Type: text/plain; charset=utf-8
  X-Openstack-Request-Id: tx0814c8d8f3694f47a13d9-0058eca7ba

Change to _id = Default:

  [filter:authtoken]
  paste.filter_factory = keystonemiddleware.auth_token:filter_factory
  auth_url = http://localhost:35357
  auth_uri = http://localhost:5000
  auth_plugin = password
  project_name = service
  username = swift
  password = Sekr3tPass
  delay_auth_decision = True
  user_domain_id = Default
  project_domain_id = Default

And:

  $ swift stat
  Account HEAD failed: http://localhost:8080/v1/KEY_fce1546e0675483bb3862a54a2ed68f6 503 Service Unavailable
  Failed Transaction ID: tx4ead4df812e2407f95b98-0058eca84d


And in the keystone log you get:

  2017-04-11 09:56:05.681 20254 INFO keystone.common.wsgi [req-cb404ba3-7762-4f8f-a84a-be8991b17851 - - - - -] POST http://localhost:35357/v3/auth/tokens
  2017-04-11 09:56:05.686 20254 WARNING keystone.auth.core [req-cb404ba3-7762-4f8f-a84a-be8991b17851 - - - - -] Could not find domain: Default.
  2017-04-11 09:56:05.687 20254 WARNING keystone.common.wsgi [req-cb404ba3-7762-4f8f-a84a-be8991b17851 - - - - -] Authorization failed. The request you have made requires authentication. from 127.0.0.1
  [pid: 20254|app: 0|req: 19/19] 127.0.0.1 () {36 vars in 536 bytes} [Tue Apr 11 09:56:05 2017] POST /v3/auth/tokens => generated 114 bytes in 7 msecs (HTTP/1.1 401) 5 headers in 225 bytes (1 switches on core 0)
  2017-04-11 09:56:13.705 20254 INFO keystone.common.wsgi [req-26eeacbe-072a-48fe-87bf-ab5e0a2053c8 - - - - -] POST http://localhost:35357/v3/auth/tokens
  2017-04-11 09:56:13.711 20254 WARNING keystone.auth.core [req-26eeacbe-072a-48fe-87bf-ab5e0a2053c8 - - - - -] Could not find domain: Default.
  2017-04-11 09:56:13.712 20254 WARNING keystone.common.wsgi [req-26eeacbe-072a-48fe-87bf-ab5e0a2053c8 - - - - -] Authorization failed. The request you have made requires authentication. from 127.0.0.1
  [pid: 20254|app: 0|req: 20/20] 127.0.0.1 () {36 vars in 536 bytes} [Tue Apr 11 09:56:13 2017] POST /v3/auth/tokens => generated 114 bytes in 9 msecs (HTTP/1.1 401) 5 headers in 225 bytes (1 switches on core 0)
  2017-04-11 09:56:13.718 20254 INFO keystone.common.wsgi [req-21127bde-4b5f-4710-a33d-1eb12a4cb5fe - - - - -] POST http://localhost:35357/v3/auth/tokens
  2017-04-11 09:56:13.723 20254 WARNING keystone.auth.core [req-21127bde-4b5f-4710-a33d-1eb12a4cb5fe - - - - -] Could not find domain: Default.
  2017-04-11 09:56:13.723 20254 WARNING keystone.common.wsgi [req-21127bde-4b5f-4710-a33d-1eb12a4cb5fe - - - - -] Authorization failed. The request you have made requires authentication. from 127.0.0.1
  [pid: 20254|app: 0|req: 21/21] 127.0.0.1 () {36 vars in 536 bytes} [Tue Apr 11 09:56:13 2017] POST /v3/auth/tokens => generated 114 bytes in 7 msecs (HTTP/1.1 401) 5 headers in 225 bytes (1 switches on core 0)

NOTING: Could not find domain: Default

So the domain_id is definitely 'default'

Now lets try name:

  [filter:authtoken]
  paste.filter_factory = keystonemiddleware.auth_token:filter_factory
  auth_url = http://localhost:35357
  auth_uri = http://localhost:5000
  auth_plugin = password
  project_name = service
  username = swift
  password = Sekr3tPass
  delay_auth_decision = True
  user_domain_name = Default
  project_domain_name = Default

And it works again:

  $ swift stat
                 Account: KEY_fce1546e0675483bb3862a54a2ed68f6
              Containers: 0
                 Objects: 0
                   Bytes: 0
         X-Put-Timestamp: 1491904775.43378
             X-Timestamp: 1491904775.43378
              X-Trans-Id: tx22ad58bb32954355acac1-0058eca906
            Content-Type: text/plain; charset=utf-8
  X-Openstack-Request-Id: tx22ad58bb32954355acac1-0058eca906

Lets finally try _name = default:

  [filter:authtoken]
  paste.filter_factory = keystonemiddleware.auth_token:filter_factory
  auth_url = http://localhost:35357
  auth_uri = http://localhost:5000
  auth_plugin = password
  project_name = service
  username = swift
  password = Sekr3tPass
  delay_auth_decision = True
  user_domain_name = default
  project_domain_name = default

And:

  $ swift stat
  Account HEAD failed: http://localhost:8080/v1/KEY_fce1546e0675483bb3862a54a2ed68f6 503 Service Unavailable
  Failed Transaction ID: txa0f514a2fdb1472bae420-0058eca984

And again the keystone logs say:

  2017-04-11 10:01:16.267 20254 INFO keystone.common.wsgi [req-9b2f788d-6950-4f57-85b5-eb9daec55e87 - - - - -] POST http://localhost:35357/v3/auth/tokens
  2017-04-11 10:01:16.272 20254 WARNING keystone.auth.core [req-9b2f788d-6950-4f57-85b5-eb9daec55e87 - - - - -] Could not find domain: default.
  2017-04-11 10:01:16.273 20254 WARNING keystone.common.wsgi [req-9b2f788d-6950-4f57-85b5-eb9daec55e87 - - - - -] Authorization failed. The request you have made requires authentication. from 127.0.0.1
  [pid: 20254|app: 0|req: 33/33] 127.0.0.1 () {36 vars in 535 bytes} [Tue Apr 11 10:01:16 2017] POST /v3/auth/tokens => generated 114 bytes in 9 msecs (HTTP/1.1 401) 5 headers in 225 bytes (1 switches on core 0)
  2017-04-11 10:01:16.280 20254 INFO keystone.common.wsgi [req-ad3103ab-7624-4554-905c-0ab14621365e - - - - -] POST http://localhost:35357/v3/auth/tokens
  2017-04-11 10:01:16.284 20254 WARNING keystone.auth.core [req-ad3103ab-7624-4554-905c-0ab14621365e - - - - -] Could not find domain: default.


Conclusion:

  *_id = default
  *_name = Default

When in doubt check keystone:

  (openstack) domain show default                                                                                     
  +-------------+--------------------+                                                                                
  | Field       | Value              |                                                                                
  +-------------+--------------------+                                                                                
  | description | The default domain |                                                                                
  | enabled     | True               |                                                                                
  | id          | default            |                                                                                
  | name        | Default            |                                                                                
  +-------------+--------------------+

project_domain_id = default
user_domain_id = default