Since you are asking about conspiracies in general and not specifically conspiracies to assassinate POTUS – that's just an example – I will draw on Washington state law. The relevant law is RCW 9A.28.040. The core of the law is that
(1) A person is guilty of criminal conspiracy when, with intent that
conduct constituting a crime be performed, he or she agrees with one
or more persons to engage in or cause the performance of such conduct,
and any one of them takes a substantial step in pursuance of such
The primary issue that must be proven at trial is that you agreed to something, not just that you discussed it. It is not a crime to talk about committing a crime, it is a crime to agree to commit a crime. The fact that the other guy took an overt step to do the deed is sufficient for a conviction, if there is an agreement.
There is a problem with your description that you "discuss the best way to murder the president, and how are are that's it's what needs to be done". As an intellectual exercise in spy-theory, one might discuss the best way to commit a crime without agreeing to do so. If, however, you are admitting that you and the others agreed that "it is what needs to be done", you would not need to utter some magic contract-language formula like "We do hereby mutually agree to undertake this task".
The notes on jury instructions for this crime elaborates on "agreement", that
all a prosecutor needs to prove is that the conspirators agreed to
undertake a criminal scheme and that they took a substantial step in
furtherance of the conspiracy.” State v. Bobic, 140 Wn.2d 250
There is ample case law showing that
An agreement to commit a crime is an essential part of a conspiracy,
State v. Miller, 131 Wn.2d 78, 929 P.2d 372 (1997), although the
agreement need not be formal. State v. Israel, 113 Wn.App. 243, 284,
54 P.3d 1218 (2002); State v. Barnes, 85 Wn.App. 638, 664, 932 P.2d
“conspiracy to commit murder by extreme indifference requires that the
conduct be intended [and] there must be an agreement (express or
implied) to engage in conduct creating a grave risk [of death], but
the result of the conduct-death-need not be intended.” In re Sandoval,
189 Wn.2d 811, 828, 408 P.3d 675
There is no finer-grained definition of "agreement" applicable in criminal cases, therefore the jury must apply their ordinary language understanding of what an "agreement" is. On the face of it, there is no reasonable interpretation whereby the defendant did not agree to carrying out the act, but perhaps there are details that are relevant that you didn't reveal.