I have been confused by the term infinite dimensional.
The first time I saw it was in the context of vector space. The way I understand it is that a function can be seen as collection of infinitely many points and thus can be seen as an infinitely long vector. Infinite here can be seen as the length of a vector, I think. https://solitaryroad.com/c850.html
However, I also know that there exist infinite dimensional vector spaces which are characterized by the property that the spaces are spanned by infinitely many basis functions. https://www.math.purdue.edu/~shao92/documents/MA265_Review_Chapter_4.pdf
The first one is referring to the length of a vector or points a function can be evaluated, whereas the second one is referring to how many functions are required to represent a specific function. Therefore, I believe they are different concept although both have the term "infinite dimension".
Is my understanding correct?
No, they're both the second one (has an infinite basis).
The solitaryroad link is also using the term in the second sense. I've never heard it used in anything like your first sense.