Sugya of Tefisa b' Safek D' Dina- Machlokes Rif and Ramban

Can someone explain the basic klalim in the Ramban, Rif, and Rosh?

From what I chap, the Ramban says that there is no tefisa in a case of safek or teiku because of taka kohen motzia miyado.

Instead, we let the one who should owe the money (the loveh or mazek) keep his money due to the principle of hamotzi michavero alav ha rayah, le safek.

He says that the case of safek bechor is conceptually akin to the case of safek mamonos; i.e. if a yisrael takes a safek bechor, we let him keep it (takfa cohen) and in cases where the tefisa, seizure of moneys, would be doubtful, we don’t allow the tefisa; tefisa in a case of safek doesn’t work.

The Rif, in Dapei haRif, 18b, cites the first mishna in HaMafkid, and says that the entire first daf of HaMafkid is all safek- if the shomer chinam doesn’t take a shevua and pays the kefel, but if the ganav then pays for the cost of the animal, the shomer chinam becomes koneh the kefel (since he paid). But at this point, the kefel is ein adam makneh davar she lo ba la olam.

The amoraim pasken this is a teiku, after many lines of raising every possible permutation and possibility, and the Rif says “cholkin,” that they’re split, because it’s a teiku. He then says, very startlingly, “de kayma lan mammon hamutal be safek cholkin.” He holds le ma’aseh that we split mamon hamutal be safek, the doubtful funds, and the bottom line is that tefisa works, despite us not knowing conclusively what the din ought to be.

The Ramban proves from Bava Metzia 6b that (A) Safek Mammon is a motzi michavero alav ha rayah and (B) Tefisa doesn’t work when there is a safek mammon, as in “takfa kohen.”

The Rif, in two fell swoops, seems to knock out the entire theory of Ramban- Rif holds that (A) tefisa does work in cases of safek mammon, and (B) safek mammon is cholkin, not motzia michavero alav ha rayah.

Ramban, though, claims that the Rif isn’t contradicting him since their two cases are dramatically different (?); the Rif’s case lacks a muchzak, while the Ramban says that the original muchzak would take possession (Ramban says, on HaMafkid, that we’re motzi the ganav; the ganav pays in beis din, and nobody ends up owning the kefel that is ein adam makneh davar she lo ba la olam.)

The Ramban responds to (A- Rif’s claim that safek mammon is cholkin, not motzia michavero alav ha rayah)by saying that the Rif is saying that we pasken hamotzi chavero alav harayah when there is the original muchzak, who is awarded the item in dispute. When there is no muchzak present, we would pasken cholkin (?).

Ramban responds to the second half of the Rif (B- Rif’s claim that tefisa works in cases of safek mammon) by saying that when there is no muchzak (this is the Rif- no muchzak), we agree with Sumchus- mammon hamutel be safek cholkin- doubtful funds are split/ cholkin. Ramban says that Sumchus makes sense, but only in the absence of muchzak (Rif). When the muchzak is present (Ramban), then we pasken hamotzi michavero alav ha ra’ayah.

So Ramban is saying that when there is no muchzak, we pasken “cholkin” but tefisa fails when there is the presence of the muchzak.

We might say that the Rif contradicts the gemara by allowing tefisa, but the Ramban says there is no stira either between the Rif and himself or the Rif and the gemara because there is no muchzak present.

To the esteemed talmidei hakhamim here: do I have the correct pshat in the machlokes Ramban-Rif on tefisa?