In some sources, Steffensen's method is the development of Newton's method to avoid computing the derivative,
- http://bit.do/Um6M
- http://cims.nyu.edu/~donev/Teaching/NMI-Fall2010/Homework4.pdf
- http://www.hrpub.org/download/201309/ujam.2013.010215.pdf
- http://folk.uib.no/ssu029/Pdf_file/Varona02.pdf
But in the other sources it's the modification of Aitken's Method,
- http://bit.do/Um6P
- http://www.math.usm.edu/lambers/mat460/fall09/lecture13.pdf
- http://bruce-shapiro.com/math481A/notes/13-Steffensen.pdf
- http://www-rohan.sdsu.edu/~jmahaffy/courses/s10/math541/lectures/pdf/week04/lecture.pdf
Is there two version of Steffensen's method?
There are not two different versions. The links you provide describe the relationship between the methods well. The methods as described are identical. The key is to envision the sequence of finite differences as a sequence as described in the derivation of Aitken's method, in other words $$p_i \stackrel{\textrm{def}}{=} \frac{f(x_{i})}{h}.$$
Then,
$$\Delta p = p_{i+1}-p_i = \frac{f(x_{i+1})-f(x_i)}{h}.$$