The two faces of $F = m*a$ As I have understood, 
$F(t)=m \cdot a(t)$
can have 2 different meanings:


*

*When applying an external force $F$ on a point mass of mass $m$, the resulting acceleration of that mass at time $t$ is $a$.

*If a point mass is going to be accelerated with $a$ (maybe by another force), the inertial force of the mass is $F$. (Pushing in the opposite direction of $a$).
Which one did Newton mean? Are these two distinct expressions, or am I thinking the wrong way?
 A: Original Newton idea was that
$$\vec{F} = m \cdot \vec{a}$$
meaning in inertial frame of reference (net or total) force equals product of mass of body and its acceleration.
Indeed, as you mentioned similar expression appears in non-inertial frame of reference
$$\vec{F'} = - m \cdot \vec{a'}$$
meaning in non-inertial frame of reference you have to add pseudo or fictitious or inertial force $\vec{F'}$ if you wish to use $\vec{F} = m \cdot \vec{a}$.
$\vec{a'}$ in second expression represents the acceleration of the non-inertial frame of reference and not the acceleration of the mass.  Also, note minus sign in the second expression, which means that pseudo force is directed in the opposite direction to acceleration of the frame of reference!
By the way, non-inertial frame of reference is usually conveniently chosen in a way that observed body is at rest (position of the body defines that frame of reference), so you end up with $\vec{a} = 0$.
A: Actually, neither of those is quite correct. The first expression is subtly wrong; the second may be either subtly wrong in the same way, or completely wrong, depending on what you mean by "inertial force."
What Newton's second law in this form really means is that the net (total) force acting on an object at time $t$ is equal to the object's mass times the object's acceleration at that time. It does not apply to any one individual force, only to the net force. For this reason it is most properly written
$$\sum \vec{F} = m\vec{a}$$
with the summation symbol to indicate the sum over all forces.
There are some restrictions on the validity of this equation; in particular, as Pygmalion pointed out, it only works in an inertial frame of reference (or can be taken to define an inertial frame of reference, in Newtonian mechanics). Also, it only works for objects whose mass is constant, and whose velocity is small. (That last one is sort of debatable depending on which definition of force you use) The more general form of Newton's second law (and in fact the way it was originally written) is
$$\sum \vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t}$$
This applies for objects of varying mass and high velocity as well.
A: Newton meant both, since action equals reaction. The first is Newton's intended interpretation--- to accelerate a body by a, you need a force equal to ma. The second is the necessary reaction force on the thing that is doing the acceleration. If you push on a body to accelerate it, the body pushes back on you in the opposite direction.
Note that the reaction doesn't care if the body is speeding up or not--- whenever you apply F units of force on a body, the reaction is there, pushing you back by an equal amount.
