For anyone interested (I doubt someone actually is), I've solved it through - basically - simulating what login page does.
// makes credentials validation, and creates IClaimsPrincipal with the acquired claims
IClaimsPrincipal principal = LoginHelper.SignIn(editEmail.Value, editPassword.Value);
// retrieves the instance of the STS (in this case my custom STS)
TrustedSecurityTokenService service = (TrustedSecurityTokenService) TrustedSecurityTokenServiceConfiguration.Current.CreateSecurityTokenService();
// creates the request manually to point to my original page (or whatever page you desire)
SignInRequestMessage request = FederatedAuthentication.WSFederationAuthenticationModule.CreateSignInRequest(Guid.NewGuid().ToString(), "http://page/i/want/to/go/after/the/validation.aspx", true);
// processes first the request...
SignInResponseMessage response = FederatedPassiveSecurityTokenServiceOperations.ProcessSignInRequest(request, principal, service);
// ...then the response is processed, and redirected to URL above
FederatedPassiveSecurityTokenServiceOperations.ProcessSignInResponse(response, Response);
This created cookies, and principal is not IsAuthenticated. As if it were process by login page (at least it seems to work so far as expected).