What is your definition of "general" here? The classic shortest path problem (which the psalm addresses) is described as finding the shortest path between two nodes, and given non-negative edge weights, Dijkstra's algorithm is perfectly general here. It will find such a shortest path in any graph, no further restrictions.

For A*, everything depends on the heuristic, as you said. If h(v) = 0, A* is equivalent to Dijkstra's algorithm, so potentially, it can be as general. But the wrong heuristic (inadmissible, inconsistent) will lead to wrong results, and so calling it a "specialized" tool is correct. The heuristic gives you a specialized version of Dijkstra's algorithm which is faster on specific graphs.

> Dinkstras is a-star with a "min weight on shortest path first" heuristic.

I am not quite sure what you mean here. Dijkstra's algorithm is A* with no heuristic. I am not sure how a "shortest path first" heuristic would look like - do you mean that Dijkstra's algorithm chooses the node to expand next based on its shortest-path cost to the target t? Even if you would construct such a heuristic h(v) = c(v, t) by explicitly calculating c(v, t) each time, this is not what Dijkstra does, as it would basically be a perfect heuristic - you would then only visit nodes on the shortest path.

If you meant that Dijkstra's algorithm is A* with a "neighboring node with shortest path from start node first" heuristic, that's also not strictly true, because expanding the nearest node first is already built-in to A*. A* does not chose the next node v with the smallest h(v), but with the smallest g(v) + h(v), where g(v) is the shortest path cost from the source node to v.