Had you tried with https:// included in the url? If not wget might have tried on the non-SSL port. If you leave out ">/dev/null 2>&1" in the command there might be more added to the cron log telling you why it fails to work (later on you should add it again to have a shorter log file). If wget still doesn't work you might want to add ">/dev/null 2>&1" also to the lynx command to suppress unnecessary output / log entries.

In addition to the "weekly" interval you might need at least the "minute" interval, too. Elgg triggers sending of notifications using the "minute" cron handler. Without a cronjob triggering this interval the notifications won't be sent. If triggering once per minute is too often for your liking (and you think users can wait longer for any notications) you can set up the "minute" cronjob at a larger interval as the important point is that the corresponding cron url gets called at all.

With the Croncheck plugin (https://elgg.org/plugins/1864611) you can see for which intervals any handlers are registered (by Elgg or a 3rd party plugin) and you would have to set up cronjobs at least for these intervals (if changing the handler to another interval isn't possible).

Dear iionly, I'm very advanced user (Drupal, WordPress).

I used all the options for the cron. 
I've domains with SSL and without SSL and still I did't get any records in the Cron statistics.
I've both minute and weekly intervals. 
In my replies here I indicate weekly only but this doesn't mean that I don't check the cron by the minute.

Of course, I tried your plugin Croncheck but it didn't bring me any good.

I believe that I made a mistake in the settings of the SSL because on my localhost everything works well.

If it matters please tell other users about this problem - use /usr/bin/lynx  instead of Elgg documentaitions.

Here's my story:

It all started on February 10.

On this day, I changed only the settings of my server.

Namely, added the following entries - Using nginx I added these records in my default.conf

ssl_protocols TLSv1.2;
ssl_ciphers 'ECDHE-ECDSA-AES256-GCM-SHA384:ECDHE-RSA-AES256-GCM-SHA384:ECDHE-ECDSA-CHACHA20-POLY1305:ECDHE-RSA-CHACHA20-POLY1305:ECDHE-ECDSA-AES256-SHA384:ECDHE-RSA-AES256-SHA384';

I checked the work of such settings on localhost and everything works well without SSL.

Therefore, I can inform you that editing DNS records can affect the work of your subdomains.

Use Elgg 3.x CLI solution instead Elgg 2.x web calls via browser.

Regards to all Elgg's team.