Can't sleep -- let's see if writing about time zones does it. :^) At least I'll clarify my sloppy use of "LMST".
Technically, I should have used HLST in the MER clocks, as I do on the tau page. LTST (local true solar time) is defined by 0:00 being local true midnight every day, with the length of a "second" (etc.) changing over the year due to the elliptical orbit (equation of time). LMST is a special case of what I call "time zone time". It has a fixed second (slightly longer than an SI second), and a fixed day (24-hour) duration that is equal to the average duration of the solar day. Once you define a meridian, you know the offset from any fixed reference time. In theory, LMST uses ones actual (local) meridian, thus it is a "solar time" that is both "local" and "mean". For MER, the actual meridian was dropped in favor of a nearby one that made LTST and planning time align during the 90-sol mission. Thus, "hybrid local solar time", HLST (not sure why they dropped "mean" and kept "local"--seems backwards). LMST is like being in the sweet spot of your time zone on Earth; HLST is like being somewhere else in your time zone. To me, that distinction is, philosophically, small. It is how you set your watch, not how fast your watch runs. If you use things like EDT or PST or CEST, you use types of HLST (even if that term is--rightly--not in general use).
In practice, I look at what NAIF says about the planning time[*] of the midnight immediately preceding landing, so I use a fixed time reference rather than a longitude. The MSL time reference is a suspiciously round number. So, I have suspected it is really an HLST (maybe just a "local-ish mean solar time"), but have never cared enough to ask or verify. I prefer to set my clock and move on. So I guess I do the mission clocks the way I use my watch on this planet--my airplane lands, they tell me what time it is; I never ask what longitude it is, and I don't adjust my watch as I "rove" until I hit a time zone line.
I'll put up a
page for Curiosity ASAP (I run the clock on my laptop). The clock won't surprise anyone who has Mars24--but you can see the numbers & math in the javascript file. (The "tau" part will be content free, probably for a long time--at this stage for MER none of that was available.)
* I use planning time as a generic term that includes HLST, LMST, or whatever framework people use when scheduling activities. LTST may factor into our thinking, but planning time uses a fixed second. I don't even want to get into SCET or SCLK[**]--those are downstream of the activity planning process. (I find that very few people know when they are with respect to seconds past Jan 1, 2000, at any given time, but non-sarcastic rovers almost always do.)
** I guess I should add: Spacecraft Event Time in UTC; Spacecraft Clock.[***]
*** Oh my, I've got footnotes to my footnotes. Time to go.