What is the definition of linear momentum?

Every where and book I search I get that the definition of linear momentum is the amount of speed (quantity of motion) contained in it or simply it is mass $\times$velocity? So, what is an appropriate definition of linear momentum? What did Newton think when he discovered it? He certainly did not think it as the amount of speed in a body.

Answer

Newton thought of momentum as "Quantity of motion" - as we can see in the translated version of 'Principia'. Particularly, he defined momentum in the following words:

The quantity of motion is the measure of the same, arise from the velocity and quantity of matter conjointly.

So yeah, that is the definition of momentum. The question why we defined the momentum the way we defined is the actual question you have in mind, I think. Well, the answer can be thought of like this.

In physics, we actually try to find some combinations of some directly measured quantities of the objects whose appropriate summation remains constant in time - no matter whatever process the system is going through. There exists different such groups of terms of such a nature that summation of terms in group remain constant in time, but individual terms do not remain contstant in time, in general. We address each term of a particular group by one name - i.e. by the name of that group. Also we assign further names to individual terms of the group to further identify them.

Like there exists a group of terms $1/2 mv^2 + mgh + ...$ which remains constant in time. We call each term of it to be the energy and then further associate different identification with each term, like Kinetic Energy, Gravitational Potential Energy, etc.

A similar other group of terms is known and we call each term of it to be some momentum. Classically we had just summation of $m$v in this group. But later on, we found that there were some other terms also in this group and all of them together remain constant in time. For example the momentum of electromagnetic fields.

So we don't priori know how and what we should define as momentum. But we observe some traits in nature and to keep a track of these patterns we define some things and associate some intuition with it. (particularly while naming. Like Newton named it as quantity of motion.)