Can't connect second external display (but only at home)
You need an active HDMI adapter.

The signal that comes out of your Mac is DisplayPort.  It has to convert the signal to HDMI to make it work.  


At work, your dock is creating that HDMI signal so in effect, it's got an active adapter built in.
At home, you've got an "HDMI to DVI" cable meaning the pins just get re-orderd, but the signal doesn't get properly recreated.  


To get the best reliability, try to avoid HDMI altogether.  It's a "consumer electronics" standard and (IMO) not up to the bar with respect for computing. 

As for connecting/converting signals, the order or reliability is as follows:


DisplayPort → DisplayPort.  It's the native signal coming out of your Mac, so no conversions mean better reliability
DisplayPort → DVI.  DVI signaling is very compatible with DisplayPort and many times an active adapter isn't necessary, though I personally still use one because I don't like dealing with these issues once I have all my cabling in place.
DisplayPort → DVI. The signals are different, especially the clock signal, which is where a number of issues arise.  You can "fix" it with a reboot which allows the signals to "sync" up.  But disconnect/reconnect and a reboot is again necessary.
HDMI → DVI.  IMO, this should be avoided, but if you must convert these signals, an active adapter is really the only way to ensure reliable connections.


Note:  These issues aren't as prevelant in Windows because the drivers for the adapters seem to have more tolerance for the signaling discrepancies.  macOS, has much less tolerance, thus the need for quality active adapters. 

If you can swing it, I would get a dock similar to the one you have at work.  It will get you the best reliability and most flexibility. 



Additional Reading


Second monitor turns blue problem
External monitors show blank screen until restart
HDMI to DisplayPort Not Working
External Monitors Flicker when plugged into my Mac