This response is based on the information provided.  We suggest you discuss with your contracting team, program manager and/or legal department as appropriate.

With regard to definitions, when the FAR does not define a term in FAR part 2 or another FAR part, "Undefined words retain their common dictionary meaning." [FAR 1.108(a)] It is best practice to have the contracting officer sign last, but signing first does not invalidate the contract.  As for confirming the signature, the offeror's proposal often identifies those authorized to sign/obligate the company.  In any event case law allows the Government to operate under the concept of apparent authority with regard to a contractor signature.  So the contractor signature, if it is of an employee of the company, will (in most cases) be legally sufficient.  This does not work in reverse.