What is the convention between setting that this organism is a strain of a specific species and what are the considerations in saying that this organism is just a strain and not a separate species entirely?
Generally strain is equivalent to sub-species or lower. but the exact usage varies depending on the type of organism. in plants or unicellular organisms it indicated all the descendants of a single specific organism, usually becasue it possesses some unique feature. Basically it is just a clade. In animals it tends to be reserved for a group of organisms that have been interbred to the point of genetic uniformity.
Almost always a strain can still interbreed with the parent population it just has identifiable feature that unites it.
Consider an example, let's say I find or manufacture a mutant wheat plant that produces much larger seeds, and all of its offspring retain those giant seeds. But I can still crossbreed it with normal wheat. That would be an example of a strain.