OK, I found the error. Thanks for the logs. It turns out that you triggered the update when system itself was updating Windows Defender, causing sort of confusion, and thus error.
Example, look at this log:
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1/6/2017 04:34:23 AM Microsoft-Windows-Windows Defender Error 2001 Maggie-PC
Windows Defender has encountered an error trying to update signatures.
New Signature Version:
Previous Signature Version: 116.72.0.0
Update Source: Microsoft Malware Protection Center
Signature Type: Network Inspection System
Update Type: Full
User: Maggie-PC\Maggie
Current Engine Version:
Previous Engine Version: 2.1.12706.0
Error code: 0x800704e8
Error description: The remote system is not available. For information about network troubleshooting, see Windows Help.
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1/6/2017 04:34:23 AM Microsoft-Windows-Windows Defender Error 2001 Maggie-PC
Windows Defender has encountered an error trying to update signatures.
New Signature Version:
Previous Signature Version: 1.233.4073.0
Update Source: Microsoft Malware Protection Center
Signature Type: AntiSpyware
Update Type: Full
User: Maggie-PC\Maggie
Current Engine Version:
Previous Engine Version: 1.1.13303.0
Error code: 0x800704e8
Error description: The remote system is not available. For information about network troubleshooting, see Windows Help.
*******************************************************************************
1/6/2017 04:34:23 AM Microsoft-Windows-Windows Defender Error 2001 Maggie-PC
Windows Defender has encountered an error trying to update signatures.
New Signature Version:
Previous Signature Version: 1.233.4073.0
Update Source: Microsoft Malware Protection Center
Signature Type: AntiVirus
Update Type: Full
User: Maggie-PC\Maggie
Current Engine Version:
Previous Engine Version: 1.1.13303.0
Error code: 0x800704e8
Error description: The remote system is not available. For information about network troubleshooting, see Windows Help.
*******************************************************************************
1/6/2017 04:34:22 AM Microsoft-Windows-Windows Defender Information 2000 Maggie-PC
Windows Defender signature version has been updated.
Current Signature Version: 1.233.4073.0
Previous Signature Version: 1.233.4026.0
Signature Type: AntiSpyware
Update Type: Delta
User: NT AUTHORITY\SYSTEM
Current Engine Version: 1.1.13303.0
Previous Engine Version: 1.1.13303.0
*******************************************************************************
1/6/2017 04:34:22 AM Microsoft-Windows-Windows Defender Information 2000 Maggie-PC
Windows Defender signature version has been updated.
Current Signature Version: 1.233.4073.0
Previous Signature Version: 1.233.4026.0
Signature Type: AntiVirus
Update Type: Delta
User: NT AUTHORITY\SYSTEM
Current Engine Version: 1.1.13303.0
Previous Engine Version: 1.1.13303.0
*******************************************************************************
1/6/2017 04:34:22 AM Microsoft-Windows-Windows Defender Information 2011 Maggie-PC
Windows Defender used Dynamic Signature Service to discard obsolete signatures.
Current Signature Version: 1.233.4073.0
Signature Type: AntiVirus
Current Engine Version: 1.1.13303.0
Dynamic Signature Type: Signature update
Persistence Path: C:\ProgramData\Microsoft\Windows Defender\Scans\RtSigs\data\09bce56bb453a93f20482f69f7db6ed44bd50839
Dynamic Signature Version: 0.0.0.0
Dynamic Signature Compilation Timestamp: 1/5/2017 8:33:55 PM
Removal Reason: Automatic
Persistence Limit Type: Duration
Persistence Limit: 216000000
*******************************************************************************
Take a look at bold text in log above. System was performing normal update operation, called Delta update (see User: NT AUTHORITY\SYSTEM). But at the very same time, User: Maggie tried to update, confusing Defender to perform a Full update, which is not desirable and thus error was presented.
You should not worry about Windows Defender. It is working normally. The only sign of worry will be definitions being older than 72 hours. As long as definitions are getting regularly updated (whether automatically or by you), Defender is fine and no signs of worry.
PS: Error code was 0x800704e8 not 0x80074e8. Caused a lot of confusion!
Prashant Kumar