There is a proposition in my book that
For a process $M_t$ to be a martingale, it is necessary that its stochastic differential $dM_t$ has no $dt$ term.
Why is this exactly? My guess is that it implies there is no change with respect to time $t$. This is speculation, and is the story told with the 2nd property given regarding conditional expectation.
A martingale only requires that: $$1. \space \forall t, E \|M_t \| < \infty$$ and $$2. \space \forall t,s>0, \space E[M_{t+s}|\mathcal{F}_t] = M_t$$