Finding the LFSR and connection polynomial for binary sequence.
This is probably just a difference in notation than any failure in
understanding or implementation. Some people define recurrence relations with subscripts in reverse order than others; the original description given by Berlekamp in his 1968 book Algebraic Coding
Theory began counting from $1$ instead of $0$ etc.  Observe that
$$x^7 + x^4 + x^3 + x^1 = x^7(1 + x^{-3} + x^{-4} + x^{-6})$$
in comparison to your $1 + x^3 + x^4 + x^6$ which you say is the
correct reciprocal of what the web site's answer should be.
So I would say that the web site seems to be following
Berlekamp's original description and giving you an answer that is "off-by-one".