after a few adjustments I would change this to:

RWLCP: Random Winner from a List of Complementing Philosophies

A hybrid design for a single winner winner systems that combines some aspects of different voting-philosophies without predictable results for strategizing. So (succesfull) strategizing would not happen much. Also importantly the randomization would be perceived by normal citizens (as non voting-experts) to be a fairly honest choice between different kind of winners which all have a fair case to say they are legitimate and have a mandate. It could also lead to a bit more humbleness in the winner instead of arrogance because the winner would know he had an element of just luck on their side.

steps:

• start with a ballot that's suited to gather all needed info
• use independ and different methods to suggest a single winner:
  1. MARS voting (Mixed Absolute Relative Score)
  2. Reciprocal Score voting
  3. SODA voting (Simple Optionally Delegated Approval)
  4. Rangevoting
  5. STAR voting
• ends by picking the ultimate winner from the combined set of suggested winners (lets make that symbolical and festive: a little schoolchild chooses blind)
• the ultimate winner is announced

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@robertpdx while not approving the 1st ranked might be preferred by probably a small group of voters, that was not my intended contribution ( I hadn't realised this). I just found the original ballot very complex visually and expected many voters would get confused or needing unnecessary amount of "cognitive load". I would expect many invalid ballots and uneasy feelings with voters about their choices and support for this method. So no change in method or tabulation intendend, just a different visual ballot-design that is less demanding on "cognitive load" for (below) average voters.

Just on ballot design: I think this would be easier for voters (less confusion and mistakes):

I would be interested in a single winner winner systems that has these basic requirements:

• start with a score ballot
• has two independ and different methods to suggest one or more winners
• if one method produces less suggested winners then the other then that method gets an extra round to add suggested winners until both methods have the same number of winners
• ends with random picking a ultimate winner from the combined set of suggested winners (lets make that symbolical and festive: a little schoolchild chooses blind)
• the winner is announced

what two methods would you propose and why?
as an example I'll give one:

1 https://electowiki.org/wiki/Smith_set and
2) https://rangevoting.org/ with the set of winners that have the maximum SUM of all scores.
This would combine some aspects of ranking and rating without predictable results for strategizing. So (succesfull) strategizing would not happen much. Also importantly the randomization would be perceived by normal citizens (as non voting-experts) to be a fairly honest choice between two different kind of winners which both have a fair case to say they are legitimate and have a mandate. It could also lead to a bit more humbleness in the winner instead of arrogance because the winner would know he had an element of just luck on their side.

on wording:

• add something like: this is intended as a single winner method
• "The candidate who wins in the most pairwise matches wins." better: "The candidate with the highest sum of points wins"

Am I right in suspecting that both in S-TM1 and S-TM2 there could be ties that are not resolved with the suggested Tie-breaking rules.
For instance for S-TM1 : when there are 20 candidates and 5 of them have the same sum of (adjusted )scores while 3 of them have also the same number of ratings [best] and 2 of them also have the same number of ratings [ worst]

what exactly do the letter S-TM stand for?

interesting!

Maybe you can compare to https://electowiki.org/wiki/MARS_voting based on this discussion: https://www.reddit.com/r/EndFPTP/comments/lil4zz/scorebetterbalance_a_proposal_to_fix_some/

also Smith//Score seems a bit simpler to me, see https://electowiki.org/wiki/Smith//Score

First off: I think you (@tec ) are on to something valuable here and I agree that sometimes the combined cost of:
1 multiple voting rounds
2 the allowance of time for deliberation, discussion, campaigning
3 the proposal of new alternatives
could lead to:
1 better collective decisions
2 more broadly shared feelings of fairness about the process
3 acceptance of the winning proposal.
4 sometimes a VSE of over 100%
What I often see lacking in mathematical models of voting systems and procedures is the recognition that feelings, irrational behaviour, learning and changing once own preferences in and after social communication are not a bug but a feature of humans and groups. So many kudos to you for your proposing Serial Approval Vote Election (SAVE)!

Then some critical remarks intended to make it clearer and better because I'm having trouble understanding both the general procedure and the tallying-rules:
a) am I right in understanding that what you call the "focused approval round" is not approval voting at all but a series of pairwise comparisons. Maybe another name for this step would be more clear?
b) I don't understand the use of Post-loop step 2, can't this be left out?
c) can you explain what happens during the procedure to the "rest"-category you mention in "My mental model of voters has us maintaining and updating rough judgement about the alternatives we know about. This set of alternatives seem good, those not so much, and the rest haven't been considered yet."
d) on any subject my preferences change when I hear and consider new arguments including other peoples preferences (and majorities or interesting minority-views) and my views become more nuanced. That would suggest a bigger scoring range would better fit my real-lifepreferences over time. Is it your view that adapting SAVE to use approval in the pre-loop, and for instance range 1..5 in the first round of the loop and range 1..10 in the second round (and keeping increasing the range in each extra round) would introduce unwanted tactical strategies?