I know that a solving a specific problem can have different runtimes on different models of computation. But can a specific algorithm have different runtimes on different models of computation?
Also, would an algorithm be considered the high level description or the implementation level description of the procedure you need to do?
Yes it can. When you take RAM model and your algorithm has a lot of multiplications (dominant operation), your multiplications take unit time. Even if the numbers are $2048$ bit numbers. But if you take multitape Turing Machine as your model, the cost of multiplication matters and will be included in complexity.
Algorithm is high level description, implementation will vary. Getting back to multiplication, algorithm describes that in one step you have to multiply two numbers, but if numbers are bigger than computer words instead of writting "*" you have to implement it.
