The confusion arises from the various meanings of the English word "execute" (even in Law!):
See: execute - definition of execute by the Free Online Dictionary, Thesaurus and Encyclopedia. - particularly items 7 and 8 below.
execute [ˈɛksɪˌkjuːt]vb (tr)
1. (Law) to put (a condemned person) to death; inflict capital punishment upon
2. to carry out; complete; perform; do to execute an order
3. to perform; accomplish; effect to execute a pirouette
4. to make or produce to execute a drawing
5. (Law) to carry into effect (a judicial sentence, the law, etc.); enforce
6. (Law) Law to comply with legal formalities in order to render (a deed, etc.) effective, as by signing, sealing, and delivering
7. (Law) to sign (a will) in the presence of witnesses and in accordance with other legal formalities
8. (Law) to carry out the terms of (a contract, will, etc.)
As you can see from the above, the signing of a will/contract and subsequently carrying out the terms of the will/contract are both covered by the term "execute".
To avoid confusion, I would rather break up the Agreement phases into:
The bidding/negotiation phase
Signature of the parties concerned confirming the Agreement
Subsequent Implementation of the Agreement.
Hope this helps