This post is longer than usual, because I am reporting back all my findings, so maybe all the tray notification issues can be resolved.
Since my latest post in this topic several days ago, I’ve been testing several conditions in relation to program update tray notifications.
First, as I already commented before, the program update tray notifications are currently being displayed, which is already good news.
About whether they are displayed more than once, my current experience is that they are displayed according to the same rules that the database updates are subject to. To be more clear, I’ll give here one of those rules, as example.
Normally, Avast checks whether there are new database updates according to the first time it gets connected to the Internet after booting, and then it repeats this check in a fixed differential time (every 4 hours is the current default, if I’m not mistaken). Avast then displays a database update tray notification, but only in the case where there is an actual database update (at least, this is the default).
So, according to the above described defaults, the PROGRAM update tray notifications are displayed every 4 hours, until the user accepts to perform the program update. In this situation, the program update tray notification appears every time, no matter whether the database is actually updated or not. This is the expected correct default behaviour of the program update tray notifications 8).
Last year, this behaviour was the same as the above described, until my program update tray notifications were not showing up AT ALL, until several days ago, when they reappeared.
BUT (there “must” be a “but” 
), the DATABASE update tray notifications are currently NOT being displayed. To clarify, I’ll take again the above example.
For “some” of the update’ checkings of Avast (those every 4 hours), the database is indeed updated, and for some, there is no new database to update. This is naturally expected. Last year, when
_ there was a new program update available; and,
_ the database was just updated,
then 2 tray notifications were displayed simultaneously; one for the usual database update, and one for the program update availability.
Currently, the tray notification for the new program update availability is indeed being displayed every 4 hours, but the database tray notification is NOT being displayed ??? ; no matter whether there was only a check, or an actual database was effectively updated.
I also changed my settings back to what they were before, with 2 shields in “off” state, and the Status Bar not monitoring those same 2 shields. I changed back those settings to my original preferences only after testing the above described behaviours with the defaults.
After changing my settings, all the above described behaviours were still the same 8). I tested booting behaviours, connecting to the Internet behaviours and frequency of updates’ checks. I found no difference.
Whether the program update tray notifications will continue “for ever”, or whether they will disappear after some time (say, 2 weeks after the first notification, or after the official first availability), or whether they disappear after some “number of notifications”, is not part of my testings. For now, they are indeed showing up.
To me, it is not YET clear, if the database updates tray notifications will appear in some future stable release together with the program update tray notifications, as they used to be displayed last year. Currently, in the context of tray notifications only and before performing the program update, there is no noticeable difference if an actual database was updated, or only the check was ran and there is no new database update.
The second issue still not resolved is if:
_ the shields settings (on/off); or,
_ the partial disabled Status Bar monitoring; or,
_ the status of “Summary → Current Status → Real-time shields:” being “off” (as if “ALL” the shields were “off”); or,
_ any other setting (in Avast program or at the severs side); or,
_ any of the above possible combinations;
are somehow interfering / influencing / changing the tray notifications behaviour.
Yes, I tested some of the possibilities. But the first time, in months, that the program update tray notification was displayed, was coincidentally after a couple of days I resetted my shield’ settings as default.
So, to resume my recent observations posted in this post:
_ The program update tray notifications are being timely displayed until I’ll accept to actually update (which I haven’t done YET);
_ I don’t know if for the next stable release, the program update tray notification(s) will be displayed again as normal/expected, while I’ll be using non-default shields’ settings;
_ No yellow exclamation mark is being displayed over the tray icon as DavidR suggested it should be until I accept the program update (I’m not saying I want the exclamation mark, or that I was actually expecting it, but that IS indeed the current default setting as he pointed out);
_ Until I decide to actually apply the new program update, only the program update tray notification is being displayed for every update check; meaning that the database update tray notification is not being displayed when an actual database is indeed being updated.
Is like if the previous database update tray notification that was being displayed “alone” during the last few months, was interchanged / replaced by the program update tray notification being displayed “alone” now, at least until I finally decide I’ll apply the program update (and I’ll still have to see/test what happens then).
Just to be sure, I haven’t applied the program update yet, so to give an additional opportunity to receive suggestions and/or feedback here to keep testing before updating, and to give some more time to Avast Team and to the Avast Servers to (maybe and hopefully):
_ display BOTH tray notifications when they BOTH should;
_ display only the program update tray notification when only that corresponds;
_ display only the database update tray notification after applying the program update and when a database update would be preformed;
_ and no tray notification should be displayed at all when only an update check is performed but not database and no program updates were found.
Of course, those are expected behaviours according to the default “update” settings.