Here's an update to my own question.
I find the sync actions and setting options in the Gmail app to be confusing. I have decided to write down my understanding of things.
I have written this from my user perspective using information gleaned from the internet and experiment with my two android devices. It may not be technically correct and perhaps it is obvious to most users but if it helps other users then great, and if there are mistakes I am happy to be told about them. Here goes!
First my understanding of the sy stem basics.
Messages for the gmail account
username@gmail.com are held on a google server. The android user views these messages via the Gmail app. The view of these messages can be filtered by assigning "labels" to individual messages. The gmail system automatically assigns the "Inbox" label to new messages which arrive in the user's gmail account. The user can change the label assigned to a message or can assign additional labels to a message. The user can create custom labels (via a web browser using the desktop version of gmail).
There is a view called "All mail" which as its name suggests is every message which is held in the google server for that user's account. Messages which have been archived ( and hence removed) from the Inbox view still exist in the All mail view but the Inbox label is removed and unless before archiving the user has assigned another label to that message, then the message will have no label.
Sync and the (confusing) use of the term in the Gmail app
From a dictionary definition, I understand that in the computing world "sync" (synchronize) means "cause (a set of data or files) to remain identical in more than one location"
So I think that for "sync" I can read " make the same as the latest change(s)"
It follows that for this to be achieved there has to be a connection between these points of more than one location, and for Gmail that connection is the internet. So any sync action will only happen when there is an internet connection between the mobile android device and Google's server.
The Sync Gmail setting option in the Gmail app achieves synchronisation of all the labels whenever there is an internet connection i.e. "sync" as the above dictionary definition.
There are also two additional setting options, Days of emails to sync and Manage labels, which use the term sync (but I think that is confusing).
These additional settings are described below. They are settings for individual labels which have been synced while there is an internet connection, but their function is to determine which of the messages of a label,
will be stored on the device thus enabling those messages to be accessed later when there is no internet connection.
Sync - Fundamental Sync Action (this is my term)
When the Gmail app is connected to the internet (and therefore to Google's gmail servers), "sync" means the dictionary definition. That is, messages are made the same as the latest date/time stamped change held on the server.
This sync action works both ways, that is, if this Gmail app generates the latest change, for example deleting a message or changing the label of messages, then these changes are communicated to the server and then out to other devices accessing that gmail account. Conversely, if a change exists on the server with a time stamp later than exists on this Gmail app, then that change is communicated from the server to the app.
This sync action is turned on in Settings/username@gmail.com/Sync Gmail.
The sync action occurs automatically when the app is started and manually when a label view window is swiped down ( I think there may be other manual ways to initiate sync).
Whenever there is an internet connection and Sync Gmail is checked ALL of the messages assigned to a label will be visible in that label view.
The additional settings, "Days of emails to sync" or "Manage labels", which appear below the Sync Gmail option DO NOT operate on this fundamental sync action with one confusing exception.
This exception is for the "All mail" view. For messages to be visible in this view, it appears to be necessary under the Manage Labels setting, to set this to None = Not synced which at the moment appears to me to be an anomaly (described later).
Additional Settings to determine Message Storage on the Device for Individual Label Views
Messages are stored in the Google server but the Days of Emails to Sync and Manage Labels settings for individual labels, gives the option to save messages on the device. This is to allow these user selected messages to be read later when there is no internet connection.
I find the wording used on the device associated with these settings to be confusing (see below)
Days of mail (emails) to sync > scroll to select from 1 to 999
This sets the Last number of days of emails which will be stored on the device if this option is selected for a label in the Manage Labels setting ( see below). It sets a common number of days for all labels.
So for " Days of emails to sync " I read " Last number of days of synced emails to store on the device "
Manage labels >
labelname > Sync messages > Sync
labelname, o None, o Last x days, o All
This setting selects the emails with the label
labelname which will be stored on the device
After selecting one of the three options and tapping OK, the screen returns to the previous page and under the Sync Messages entry there is a (confusing) description of that selected option as follows in italics ;
Sync set to None :
No messages will download
For this I read "No synced messages will be stored on the device"
Sync set to Last x days :
Last x days of messages will download when connected (phone)
The previous x days of messages will download when connected (tablet)
For this I read " The last x days of synced messages will be stored on the device"
Sync set to All :
All messages will download when connected
For this I read " All synced messages will be stored on the device"
Going back one screen further i.e. to the Manage labels screen, each label will have a description under it depending on the option selected as in italics below ;
None :
Not synced
Last x days :
Last x days
All : All
All mail view exception
As previously mentioned these additional settings do not alter the sync action which occurs when there is an internet connection. Irrespective of these settings all messages of a label will be shown in its view.
The exception is All mail. When connected to the internet the All mail view behaves as follows ;
Sync set to None. All messages are displayed in the view
Sync set to 8 days. All messages from day 9 and earlier days are displayed in the view ( x = 8 in this example)
Sync set to All. No messages at all are displayed in the view
These settings do not achieve any storage of messages for the All mail view that I can make happen.
Finally I've noticed that if the Inbox view is set to None, all messages will be displayed when there is an internet connection and at the top of the view there is the message " Inbox sync is off. Turn it on in Inbox settings".
As previously mentioned that message means "storage of synced Inbox messages on the device is off ".
That's it, I hope it makes some sort of sense.
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