Dumb question- Hostnames

I have a lot of ESP devices on my net, most are running Tasmota and controlled with Home Assistant.

The Tasmota information screen shows my hostname, for example, as “relay1.local”, but when I do a trace route on relay1.local, the response says the hostname is “relay1.home”.

C:\Users\steve>tracert relay1.local
Tracing route to relay1.local [192.168.1.11]
  1   242 ms     1 ms     3 ms  relay1.home [192.168.1.11]
Trace complete.

Traceroute on “relay1.home” results in the same, but significantly faster:

C:\Users\steve>tracert relay1.home
Tracing route to relay1.home [192.168.1.11]
  1     1 ms     1 ms    17 ms  relay1.home [192.168.1.11]
Trace complete.

Tracing the IP address is also quite a bit faster than the .local:

C:\Users\steve>tracert 192.168.1.11
Tracing route to relay1.home [192.168.1.11]
  1     2 ms     1 ms     1 ms  relay1.home [192.168.1.11]
Trace complete.

So here’s a couple of questions.

  1. Why does the Tasmota information screen say the hostname is “relay1.local” while a ping, tracert or IP scan all say it is “relay1.home”?
  2. Does it matter? The ping times are about the same for all three addresses for “relay1” while the tracert of “relay1.local” is usually longer.