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In my research to Reversis and the Strassburger game 1637 I detected "Losing Lodam", a game, which falls in the trick-avoiding category. It's English and from 16th century.
http://www.davpar.eu/histocs/losinglo.html
The tricks in this game are not fixed in their ending as usual in rounds, but proceed till a specific state is reached: Till somebody neither can increase the cards of the played suit and trump this suit higher than the preceding player.
In this game the cards are laid on one spot, so they are not thrown in the middle as usual in trick games.
In the first Latimer text: "Now turn up your trump, your heart (hearts is trump, as I said before), and cast your trump, your heart, on this card; and upon this card you shall learn what Christ requireth of a christian man, not to be angry, ne moved to ire against his neighbour, in mind, countenance, nor other ways, by word or deed. Then take up this card with your heart, and lay them together: that done, you have won the game of the Turk, whereby you have defaced and overcome him by true and lawful play."
In the second Latimer text: "What Christ requireth of a christian man? By and by cast down your trump, your heart, and look first of one card, then of another. The first card telleth thee, thou shalt not kill, thou shalt not be angry, thou shalt not be out of patience. This done, thou shalt look if there be any more cards to take up; and if thou look well, thou shalt see another card of the same suit, wherein thou shalt know that thou art bound to reconcile thy neighbour. Then cast thy trump upon them both, and gather them all three together, and do according to the virtue of thy cards; and surely thou shalt not lose. Thou shalt first kill the great Turks, and discomfort and thrust them down. Thou shalt again fetch home Christ's sheep that thou hast lost; whereby thou mayest go both patiently and with a quiet mind unto the church, and make thy oblation unto God; and then, without doubt, he will hear thee.
Both passages possibly indicate a game mechanism, which isn't the standard for trick taking games. The second passage seems to say, that one should help the neighbor with the own cards.
If we would assume, that the game is a puzzle ... and the aim is to complete the puzzle in a right way. This would be a game, in which everybody could win. As a possible theme of the puzzle (the Turk is twice mentioned in both passages) one might suspect, that the aim is to find the inner balance in the Christian communities to resist the danger of the outside. The cards form a great picture with some ideological content.
... perhaps something like the oracle of Seni ...

These cards have 4 views on one card, and all together are a great picture (6x6 cards).
Lamiter said about his deck, that the cards had 4 parts: "Now behold and see, this card is divided into four parts".
Well, I don't claim, that the Lamiter's was like this, but perhaps one should change the general expectation to get this point right. My idea is, that it wasn't the usual trick-taking game.
The Losing-Lodam game finally ends its tricks finally always with a strong card (trump). Something like this is also possible. According Latimer's description finally everything is governed by hearts, if one plays the game in the manner, that Latimer intended.