Answer 1

No, the description sounds very different from the meaning of "Depth-first search".

The first paragraph from here says it correctly:
https://en.wikipedia.org/wiki/Depth-first_search
I'll explain more since that paragraph is likely not clearing it up for you.

A depth-first search visits or traverses through a tree to the deepest node before backtracking and visiting other nodes.

Depth-first search is in contrast to breadth-first search that looks at all shallowest nodes before visiting deeper.

Depth-first and breadth-first traversal is another common phrase that basically means the above but might be traversing for other purposes than search. It could be traversing to print every node, build a list of all values in a tree...

Something that might confuse you while reading about this elsewhere is references to "graphs". I'm explaining with reference to a tree but some articles may say "connected component" or "graph". A tree is a kind of graph. A tree has a root node. When you're doing a depth-first or breadth-first search, you need to start at a certain node that could be treated as root of a tree. It is accurate to say that the search is done on a tree even if that tree is part of a larger graph. For depth-first, it simply won't work for anything but a tree. If a node cycles back to a previously visited node, your depth first search will loop infinitely in that cycle.