If you consider them as laws, then there must be independent definitions of force and mass but I don't think there's such definitions.

If you consider them as definitions, then why are they still called laws?

In my view, standard statements of Newton's laws are usually overly concise, and this lack of detail causes confusion about what is a definition, and what is an empirical fact.  To avoid this confusion, let's proceed in a systematic way that makes the distinctions between these definitions and empirical statements clear.
What follows certainly is not the original statement of the laws made by Newton himself; it is a modern interpretation meant to clarify the foundations of Newtonian mechanics.  As a result, the laws will be presented out of order in the interest of logical clarity.
To start off, we note that the definitions of mass and force given below will require the concept of a local inertial frame.  These are frames of reference in which when an object is isolated from all other matter, it's local acceleration is zero.  It is an empirical fact that such frames exist, and we'll take this as the first law:
First Law. Local inertial reference frames exist.
How is this in any way related to the first law we know and love?  Well, the way it is often stated, it basically says "if an object isn't interacting with anything, then it won't accelerate."  Of course, this is not entirely correct since there are reference frames (non-inertial ones) in which this statement breaks down.  You could then say, all right, all we need to do then is to qualify this statement of the first law by saying "provided we are making observations in an inertial frame, an object that doesn't interact with anything won't accelerate," but one could then object that this merely follows from the definition of inertial frames, so it has no physical content.  However, going one step further, we see that it's not at all clear a priori that inertial frames even exist, so the assertion that they do exist does have (deep) physical content.  In fact, it seems to me that this existence statement is kind of the essence of how the first law should be thought because it basically is saying that there are these special frames in the real world, and if your are observing an isolated object in one of these frames, then it won't accelerate just as Newton says.  This version of the first law also avoids the usual criticism that the first law trivially follows from the second law.
Equipped with the first law as stated above, we can now define mass.  In doing so, we'll find it useful to have another physical fact.
Third Law.  If two objects, sufficiently isolated from interactions with other objects, are observed in a local inertial frame, then their accelerations will be opposite in direction, and the ratio of their accelerations will be constant.
How is this related to the usual statement of the third law?  Well, thinking a bit "meta" here to use terms that we haven't defined yet, note that the way the third law is usually stated is "when objects interact in an inertial frame, they exert forces on each other that are equal in magnitude, but opposite in direction."  If you couple this with the second law, then you obtain that the product of their respective masses and accelerations are equal up to sign; $m_1\mathbf a_1 = -m_2\mathbf a_2$.  The statement of the third law given in this treatment is equivalent to this, but it's just a way of saying it that avoids referring to the concepts of force and mass which we have not yet defined.
Now, we use the third law to define mass.  Let two objects $O_0$ and $O_1$ be given, and suppose that they are being observed from a local inertial frame.  By the third law above, the ratio of their accelerations is some constant $c_{01}$;
\begin{align}
  \frac{a_0}{a_1} = c_{01}
\end{align}
We define object $O_0$ to have mass $m_0$ (whatever value we wish, like 1 for example if we want the reference object to be our unit mass), and we define the mass of $O_1$ to be
\begin{align}
  m_1=-c_{01}m_0
\end{align}
In this way, every object's mass is defined in terms of the reference mass.
We are now ready to define force.  Suppose that we observe an object $O$ of mass $m$ from a local inertial frame, and suppose that it is not isolated; it is exposed to some interaction $I$ to which we would like to associate a "force."  We observe that in the presence of only this interaction, the mass $m$ accelerates, and we define the force $\mathbf F_{I}$ exerted by $I$ on $O$ to be the product of the object's mass and its observed acceleration $\mathbf a$;
\begin{align}
  \mathbf F_{I} \equiv m\mathbf a
\end{align}
In other words, we are defining the force exerted by a single interaction $I$ on some object of mass $m$ as the mass times acceleration that a given object would have if it were exposed only to that interaction in a local inertial frame.
Second Law. If an object $O$ of mass $m$ in a local inertial frame simultaneously experiences interactions $I_1, \dots, I_N$, and if $\mathbf F_{I_i}$ is the force that would be exerted on $O$ by $I_i$ if it were the only interaction, then the acceleration $\mathbf a$ of $O$ will satisfy the following equation:
\begin{align}
  \mathbf F_{I_1} + \cdots +\mathbf F_{I_N} = m \mathbf a
\end{align}
