If you're interested in astronomical observations, for example of satellites such as GPS, or if you want to cite a technical standard (ITU-R TF.460-6), then you might care that we use UTC and that GMT no longer has a precise definition. Otherwise you probably consider GMT to be the same thing as UTC, and also consider UT and UT1 to be the same as UTC — which technically they are not.
Also, if you're tracking computer criminals or other distributed activity, then you need to determine whether certain events at various sites may or may not have occurred before certain other events. For that purpose you will want to learn and use Network Time Protocol (NTP). That will have a much bigger effect on your understanding of time than the little differences between UTC, UT1, and UT.
"International Date Line West" is just a friendly name for a timezone where the time is defined as twelve hours less than UTC (that is, UTC-12).
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