Is the *greedy* best-first search algorithm different from the best-first search algorithm?

The __wiki page__ has a separate paragraph about Greedy BFS but it's a little unclear.

My understanding is that Greedy BFS is just BFS where the "best node from OPEN" in Wikipedia's algorithm is a heuristic function one calculates for a node. So implementing this:

OPEN = [initial state] CLOSED = [] while OPEN is not empty do

1. Remove the best node from OPEN, call it n, add it to CLOSED.

2. If n is the goal state, backtrace path to n (through recorded parents) and return path.

3. Create n's successors.

4. For each successor do:

a. If it is not CLOSED: evaluate it, add it to OPEN, and record its parent.

b. Otherwise: change the recorded parent if this new path is better than the previous one.

done

with "best node from OPEN" being a heuristic function estimating how close the node is to the goal, is actually Greedy BFS. Am I right?

*EDIT:* Comment on Anonymouse's answer:

So essentially a greedy BFS doesn't need an "OPEN list" and should base its decisions only on the current node? Is this algorithm GBFS:

1. Set START as the CURRENT node

2. Add CURRENT to Path [and optionally, to CLOSED?]

3. If CURRENT is GOAL, exit

4. Evaluate CURRENT's successors

5. Set BEST successor as CURRENT and go to 2.