There is an antagonism between specificity (which generally improves usefulness) and generality (another term for non-specificity which always improves probability under uncertainty). The logical relation is
implication. "P implies Q" means that P is at least as specific as Q. (P cannot be true unless Q is, too.)
Example: "Today's high temperature will be 32 degrees C" is potentially more useful than "Today will be a hot one." However, so long as the matter is uncertain, the latter statement is more plausible. If the high temp turns out to be 31C or 33C, the more specific statement is false, but the less specific one is still true.
Under all normative definitions of plausibility if P implies Q, then Q is no less plausible than P.
Under Bayes and similar plausibility=probability models, if uncertain P implies Q, but Q doesn't imply P, then Q is more probable than P.
Andrew (like everybody else) probably wants plenty of both: high specificity and high plausibility. Sure, no doubt there's a "sweet spot" where whatever the loss in plausibility in specifying a reason for Jesus having followers is offset by the improved usefulness of "People followed Jesus because he worked miracles" compared with "People followed Jesus."
"Minimal" as Carrier uses the term in the phrase
minimal historical Jesus, means most general (= least specific) provided we require some level of usefulness. Andrew may not find Carrier's chosen minimal level of usefulness interesting, which is fine. The posted price is to give up some plausibility for each and every gain in specificity.
What then is the relationship between Carrier MHJ (C-MHJ for short):
1. An actual man at some point named Jesus acquired followers in life who continued as an identifiable movement after his death.
2. This is the same Jesus who was claimed by some of his followers to have been executed by the Jewish or Roman authorities.
3. This is the same Jesus some of whose followers soon began worshiping as a living god (or demigod).
and what I take to be your proposal's formal definition (hereafter D-MHJ):
1. He was a humble man. He came with no supernatural power, and therefore didn't perform miracles. He lived a life devoted to the Law.
2. He became a martyr, by challenging either the Romans or the Jews over some point of Law that he decided to adhere to, even though he knew he'd suffer for it
3. He was crucified
4. Visions of him after his death led to his identification as the Suffering Servant in Isaiah and also the Christ
5. Once he was identified as the Christ, people starting searching the Scriptures to create stories about him. This rapid legendary development led to the writing of the Gospels
The logical relationship between C-MHJ and D-MHJ is nuanced. D-MHJ seems to imply that Jesus was a real man (that much of C-MHJ 1), and apparently acquired followers (somebody had the visions in 4 and did the identifying in 5). However, there is nothing in D-MHJ about him ever being called Jesus nor acquiring followers in life nor their having anything to do with an identifiable movement.
Here's what I don't think
I don't think you intended to be more specific than C-MHJ on some points (Suffering Servant) but less specific than Carrier on others. In particular, I don't think you dispute the Carrier points which you omit. I think you just simply overlooked these items, and that the following would be a fair representation of your actual position (mark it X-MHJ for identification):
1. An actual man at some point named Jesus acquired followers in life who continued as an identifiable movement after his death. He was a humble man. He came with no supernatural power, and therefore didn't perform miracles. He lived a life devoted to the Law.
2. This is the same Jesus who was claimed by some of his followers to have been executed by the Jewish or Roman authorities. He became a martyr, by challenging either the Romans or the Jews over some point of Law that he decided to adhere to, even though he knew he'd suffer for it. He was crucified.
3. This is the same Jesus some of whose followers soon began worshiping as a living god (or demigod). Visions of him after his death led to his identification as the Suffering Servant in Isaiah and also the Christ. Once he was identified as the Christ, people starting searching the Scriptures to create stories about him. This rapid legendary development led to the writing of the Gospels.
X-MHJ implies both D-MHJ and C-MHJ. It is therefore less probable than either one, but more specific than either one.
More useful than C-MHJ? For some purposes, yes - Andrew and you are entitled to your 'druthers in the unavoidable trade-off between plausibility and specificity - but Carrier had a specific usage in mind.
C-MHJ stands in for all the more specific relevant HJ's in order that its defeat in match play against a minimal Doherty-Carrier hypothesis means that each and every HJ is less plausible than simple Doherty-Carrier. All the relevant HJ's taken together are assessed less likely true than not.
I don't see that Carrier can be criticized for that decision within a Bayesian framework. Nor does your preference for more specificity make his trade-off "wrong" for the particular purpose he chose. I think there is a real problem where your preferred MHJ, with its trade-off between specificity and plausibility for whatever purpose, fits into a discussion of Carrier's design of a trade-off to suit a specified purpose.
I've tried to stick to the earliest texts as much as possible, and from them I think I've built a consistent picture, though admittedly one built off minimal data points. One can draw anything from a small number of dots.