Almost all sources of information quoting the Legendre Polynomials includes the following forms (I have included the first 4 polynomials):
$P_0(x) = 1 $
$P_1(x) = x $
$P_2(x) = 1/2(3x^2-1) $
$P_3(x) = 1/2(5x^3-3x) $
$P_4(x) = 1/8(35x^4-30x^2+3) $
However, in their textbook Numerical Analysis 10E by Burden et al, they quote the Legendre Polynomials as:
$P_0(x) = 1 $
$P_1(x) = x $
$P_2(x) = x^2-1/3 $
$P_3(x) = x^3-3x/5 $
$P_4(x) = x^4-6/7x^2+3/35 $
I understand that the second form above is essentially the first form equated to zero and dividing out the highest power co-efficient. Is there a reason as to the two differing forms, do they have a different name and in what context is one used over the other?
When it comes to sequences of polynomials or even rational numbers, there can be reasons to use one form rather than another. One good example is the Bernoulli number sequence. At the beginning of the Wikipedia article it mentions two almost identical variations related by $\,B_n^+ = (-1)^n B_n^-.\,$ which only differ in the value of $B_1.\,$ Another variation uses the notation $\,B_n := B_{2n}\,$ which has the advantage that all of the numbers are non-zero. For this see the MathWorld Bernoulli number article. Still another variation of Bernoulli numbers is the Tangent numbers given by $\,T_n = 2^{2n}(2^{2n}-1)|B_{2n}|/(2n)\,$ which has the advantage of being an integer sequence.
Similar things happen with polynomial sequences such as the Legendre polynomials. The polynomials can be normalized in a few ways. For example, given an inner product of some kind, we can normalize them to have norm 1. Of course this depends on the inner product chosen and the article mentioned the "shifted" Legendre polynomials that arise in this way which has the advantage that all the coefficients are integers and $P(0)=1$. We can also normalize by requiring $P(1)=1$. The textbook in your question chose to normalize by dividing by the leading coefficient to turn them into monic polynomials. I am not sure what advantage this has other than being monic.