Cheaply repair burnt / melted vinyl floor
If these are minor holes that don't cause loss of function of the floor, I think they are probably "wear and tear" that is a legally mandated part of every rental agreement (in the USA), so you could not worry about it and leave them be...

Per your question, properly glued this is way the professionals patch.  I have noticed that they tend to put a little fillet of glue around the border if the patch, I suppose so you are less likely to catch the edge and rip it back up. I'm not sure what the glue is - sorry.

Cut the patch piece first (in rectangular shape) then lay it over the bad spot and use it as a guide to cut the damaged flooring.  This will help you get a patch that's the exact size for the hole.