Cron daily running twice

On my CentOS 7 server, errors generated by a nightly backup script that should have run fine caused me to check on my cron activity. I discovered that cron.daily is running twice – here is the relevant section of /var/log/cron after I removed my backup script to see if it was somehow causing the problem:

Oct 10 02:28:01 mail CROND[1750]: (root) CMD (run-parts /etc/cron.hourly)
Oct 10 02:28:01 mail run-parts(/etc/cron.hourly)[1750]: starting 0anacron
Oct 10 02:28:01 mail anacron[1759]: Anacron started on 2017-10-10
Oct 10 02:28:01 mail run-parts(/etc/cron.hourly)[1761]: finished 0anacron
Oct 10 02:28:01 mail anacron[1759]: Normal exit (0 jobs run)
Oct 10 02:30:01 mail CROND[1766]: (root) CMD (/usr/lib64/sa/sa1 1 1)
Oct 10 02:40:01 mail CROND[1847]: (root) CMD (/usr/lib64/sa/sa1 1 1)
Oct 10 02:50:01 mail CROND[1936]: (root) CMD (/usr/lib64/sa/sa1 1 1)
Oct 10 03:00:01 mail CROND[2032]: (root) CMD (/usr/lib64/sa/sa1 1 1)
Oct 10 03:10:01 mail CROND[2148]: (root) CMD (/usr/lib64/sa/sa1 1 1)
Oct 10 03:17:01 mail CROND[2223]: (root) CMD (run-parts /etc/cron.daily)
Oct 10 03:17:01 mail run-parts(/etc/cron.daily)[2223]: starting kizunademo
Oct 10 03:17:02 mail run-parts(/etc/cron.daily)[2259]: finished kizunademo
Oct 10 03:17:02 mail run-parts(/etc/cron.daily)[2223]: starting logrotate
Oct 10 03:17:02 mail run-parts(/etc/cron.daily)[2266]: finished logrotate
Oct 10 03:17:02 mail run-parts(/etc/cron.daily)[2223]: starting man-db.cron
Oct 10 03:17:02 mail run-parts(/etc/cron.daily)[2277]: finished man-db.cron
Oct 10 03:20:01 mail CROND[2288]: (root) CMD (/usr/lib64/sa/sa1 1 1)
Oct 10 03:28:01 mail CROND[2367]: (root) CMD (run-parts /etc/cron.hourly)
Oct 10 03:28:01 mail run-parts(/etc/cron.hourly)[2367]: starting 0anacron
Oct 10 03:28:01 mail anacron[2376]: Anacron started on 2017-10-10
Oct 10 03:28:01 mail run-parts(/etc/cron.hourly)[2378]: finished 0anacron
Oct 10 03:28:01 mail anacron[2376]: Will run job `cron.daily' in 35 min.
Oct 10 03:28:01 mail anacron[2376]: Jobs will be executed sequentially
Oct 10 03:30:01 mail CROND[2381]: (root) CMD (/usr/lib64/sa/sa1 1 1)
Oct 10 03:40:01 mail CROND[2462]: (root) CMD (/usr/lib64/sa/sa1 1 1)
Oct 10 03:50:02 mail CROND[2547]: (root) CMD (/usr/lib64/sa/sa1 1 1)
Oct 10 04:00:01 mail CROND[2670]: (root) CMD (/usr/lib64/sa/sa1 1 1)
Oct 10 04:03:01 mail anacron[2376]: Job `cron.daily' started
Oct 10 04:03:01 mail run-parts(/etc/cron.daily)[2685]: starting kizunademo
Oct 10 04:03:02 mail run-parts(/etc/cron.daily)[2721]: finished kizunademo
Oct 10 04:03:02 mail run-parts(/etc/cron.daily)[2685]: starting logrotate
Oct 10 04:03:02 mail run-parts(/etc/cron.daily)[2728]: finished logrotate
Oct 10 04:03:02 mail run-parts(/etc/cron.daily)[2685]: starting man-db.cron
Oct 10 04:03:03 mail run-parts(/etc/cron.daily)[2739]: finished man-db.cron
Oct 10 04:03:03 mail anacron[2376]: Job `cron.daily' terminated
Oct 10 04:03:03 mail anacron[2376]: Normal exit (1 job run)

Why is cron.daily running twice? As you can see, the log contains some entries related to the second run that aren't present for the first run: two lines announcing the upcoming run, and two more lines saying that it terminated with a normal exit. The first run simply ran the scripts with no extra fanfare. I assume that means something, but I don't know what.

I checked everything I could think of for doubles of something. I'm pretty sure I've read every similar thread on the subject, so compare with the following before calling this a duplicate question. Based on Why is cron running twice? I checked for extra processes – the complete output of ps aux | grep cron is as follows, so there is only one process:

root      9383  0.0  0.2 112672  2340 pts/0    S+   15:18   0:00 grep --color=auto cron
root     25624  0.0  0.0 126248   320 ?        Ss   Sep30   0:02 /usr/sbin/crond -n

Based on Cron jobs running twice – Ubuntu server 12.04 I also checked crontab -l -u root, which said no crontab for root.

And here is my /etc/crontab:

SHELL=/bin/bash
PATH=/sbin:/bin:/usr/sbin:/usr/bin
MAILTO=my@email.com

28 * * * * root run-parts /etc/cron.hourly
17 3 * * * root run-parts /etc/cron.daily
44 2 * * 0 root run-parts /etc/cron.weekly
8 2 7 * * root run-parts /etc/cron.monthly

Thoughts?

EDIT (9 months after this discussion had fallen silent):

Comment today from Marin Velikov made me aware that there is an anacrontab file (I know it's silly, but it hadn't even occurred to me). Here is its contents:

SHELL=/bin/sh
PATH=/sbin:/bin:/usr/sbin:/usr/bin
MAILTO=root
# the maximal random delay added to the base delay of the jobs
RANDOM_DELAY=45
# the jobs will be started during the following hours only
START_HOURS_RANGE=3-22

#period in days   delay in minutes   job-identifier   command
1       5       cron.daily              nice run-parts /etc/cron.daily
7       25      cron.weekly             nice run-parts /etc/cron.weekly
@monthly 45     cron.monthly            nice run-parts /etc/cron.monthly

So that's the cause. But why would the system be configured to run both? I assume someone smarter than me set it up this way, so I hesitate to muck with it before knowing the purpose. And if I should indeed get rid of the entries in either crontab or anacrontab, which one is best? Anacron is apparently the more sophisticated tool, but it seems weird/wrong to empty one's crontab. Am I just too old-school?