In Classical Latin, did the prefix 'de-' in 'deputare' mean anything semantically?

Question

deputy (n.)     c. 1400, "one given the full power of an officer without holding the office," from Anglo-French deputé,
noun use of past participle of Middle French députer "appoint, assign" (14c.),
[3.] from Late Latin deputare "to destine, allot,"
[2.] in classical Latin "to esteem, consider, consider as," literally "to cut off, prune,"
[1.] from de- "away" (see de-) + putare "to think, count, consider," literally "to cut, prune" (see pave).

This answer on Linguistics SE helpfully explains the Semantic Field of 'putare'.

4. Wiktionary's entry states the same meanings as Etymonline above: Am I correct to interpret that the prefix de- added nothing semantically to the semantic shift from 1 to 2: putare alone already means the meanings in 2?

5. The entitled question concerns only 1 vs. 2 though; I do conjecture de- to add semantically to the Semantic Shift from 2 to 3, because destining or alloting something can be interpreted as cutting or pruning something, and then delegating what was cut or pruned to someone else.

Answer 1

Entry on Lewis and Short's dictionary. Here are the relevant senses, in my opinion:

I. the going out, departure, removal, or separating of an object from any fixed point.

C. In other relations, implying separation, departure from, etc.

1. To designate the whole, from which a part is taken, or of which a part is separately regarded, etc., from among, out of, from

II. In composition the e becomes short before a vowel

2. Signif.

   a. Separation, departure, removal, taking away; off, away, down, out: decedo, demigro, demeto, depromo, descendo, devolvo, derivo, deflecto, etc.; and trop. dedico, denuntio; and in a downward direction, decido, decumbo, deprimo, demergo, delabor, defluo, demitto, desido, desideo, declivis, deculco, degredior, deicio, etc.

Essentially, the sense is "down from," supposedly because the deputation is made by a magistrate transferring powers "down" to the deputy.