Lanthanum shouldn't show up at all, since the way it works is by binding to po4 and forming precipitate. It should do this within the skimmer chamber and get skimmed out, so it makes sense not to see it. If it shows up it means my po4 has bottomed out and the lanthanum is free in the water column and/or I majorly overdosed it (see below).
I actually did a FM and Triton ICP at the same time a few months back - the results were pretty close. I was so annoyed that the FM test did not include a shipping label that I wouldn't do it again. Triton was faster too and I prefer the way they report the data (this is subjective of course). I actually sent a 3rd brand too (I think it was reef lab something?).
While I don't think any ICP is spot on, especially for traces, I have grown to trust Triton somewhat. When I had a lanthanum overdose event while accidentally having the skimmer off, it immediately showed up on the next ICP. I also see my iodine results move based on very predictable actions - the increase/decrease of the high-iodine captiv8 product and the increase/decrease of my nori feeding. Po4 is always pretty close to my Hanna (though it reads low, given the bio processes that continue occurring in the test tube). It also seemed to roughly reflect the corrective magnesium dosing I finally had to do a few weeks back (first time ever dosing Mg).
Basically, the results generally reflect what I would expect based on the actions I've taken. So while I don't consider any of it gospel, it's trustworthy enough for my needs. Which is mostly just making sure nothing is too out-of-whack. I don't really intend to do any moonshiner style trace dosing as mentioned above.
To your point about color, I *MAY* experiment with some of the more recognized trace elements (like Flouride) but it would just be supplemental, not necessarily a targeted approach.
It would be great if the skimmer could skim out the precipitate entirely. One of the main reservations I still have about using LC is how it resolves high PO4 issues through precipitation - which is very similar to Kalkwasser (and my opinion on this is known). Yet, it is still much better than, e.g., GFO and can be very precisely dialed in. I know you have the bigger version of the skimmer I have and love how well it runs, so if this is the case, I might be inclined to use lanthanum in my other tank, too, if I have to. I used lanthanum on a couple of other skimmers, including the Nyos 120, and it could be measured through ICP in the water despite PO4 being very detectable.
It's good to know the results were close - I will comment on the FM ICP in my journal to not clutter your journal, and since I am currently working through my results, in a nutshell, Triton is more convenient for shipping but also more expensive. Triton's report is not great for folks who do not understand how to read an ICP report and might take wrong actions from it. Showing 0 values as acceptable on several traces is not good advice, in my opinion. While some may not need to be detectable (and are not) to be adequate, others should be seen to be sufficient. So, determining what is 'out of whack' is somewhat biased through the way they report this.
I also like that your results about the Hanna testers seem to match, minus the shipping time. If you use them properly, these home testers are not as bad as I read and saw people talking about them often.
Keith/Reef Bum (also an ICP consultant for FM, so maybe somewhat biased by this) restated in the last show recently how his acros colors have so much improved once he dialed in his traces to a more precise level. If this is worth the effort, though, is the question, but dialed-in traces might not only improve colors but long-term coral health and resistance to diseases and parasites. It is a fun and interesting part of our hobby the more you get into this.