I apologize if I'm not exactly answering your question, as I can't speak directly to Yoshinkan or Shodokan: I am not a part of either organization.
However, I am a Hapkido-in, and who practices the equivalent tegatana as Yoshinkan; and I am an Aikido-ka with Aikikai, whose practice of tegatana is similar in Shodokan according to your descriptions.
So I would be offering only a guess based on experience from different styles.
My observation is that neither statement you made is entirely correct, unless the 4 styles hold tegatana in 4 different ways.
In Hapkido, we splay the fingers, not because the hand is relaxed, but because that allows energy - they say "ki" - to reach the hand as it exits the fingers. This makes the hand very tense. That protects the hand; in Hapkido, we often strike with such a hand, and so it is necessary that the hand is tense: and that is the point, the hand is not relaxed. This is different in your observation of tegatana in Yoshinkan, where the splayed fingers relax the hand. In an attempt to get my hand in the picture you provided, I find my hand is far from relaxed. It is tense, not unlike how I might do it in Hapkido. Therefore, it's possible I am not doing it according to that style's guidelines.
In Aikikai, the fingers aren't firmly together, but they aren't splayed, either. In fact, keeping the fingers together has the tendency to tense the hand, just as splaying the fingers does the same. I think, then, that is a sort of "middle road", because in this case, the hand is very relaxed. This makes sense: we tend not to strike when the hand is in this position. A tensed hand can give away our intentions and location - even if subconsciously - as we feel for the moment to take kotegaishi or assist in iriminage or something.
In Aikikai, there is the desire to not want to force anything, and so, a tensed hand would violate that policy. In Hapkido, we do a lot of striking, and it makes sense that the hand is tensed.
If the hand is tense, then there is the desire for that hand to become "alive" and actively part of a technique, whereas a "dead" hand is more passive. In my Aikikai school, it is not uncommon for the same hand to shift from passive (dead) to active (alive) during the course of a technique. Kotegaishi is a good example (and one which we perform the same technique in Hapkido): the passive hand feels down from shoulder to upper arm to lower arm; and at the point we feel the wrist, we grab the wrist according to our style's guidelines (which differs from style to style, and instructor to instructor, so, I'll leave that part out).
But whether these principles are the same or similar in Yoshinkan or Shodokan, I can't say.