bjt biasing and applying input signal
Question: Now, if I add the coupling capacitor in the second one does that make the two figures equivalent? @LvW

No - they are not.
In the first circuit we have an IDEAL dc source (stiff), whereas in the 2nd circuit we have a voltage divider which delivers a voltage across a finite source resistance. As a consequence, the dc bias voltage at the base is not independent on the current into the base (which depends on the particular part and the temperature). However, because of practical reasons the 2nd circuit is, of course, used (with a current stabilizing resistor in the emitter path).