How the summary is picked
This uses extractive summarization: it scores every sentence by how many frequently-repeated, meaningful words it contains (common filler words like "the" and "and" are ignored), then picks the highest-scoring sentences and puts them back in their original order as bullet points. It is the same family of technique used by many "TL;DR" and auto-summary tools - no sentence is rewritten, each bullet is an actual sentence lifted from the source.
Why pasted text and a URL are handled by two different processes
Pasted text is summarized entirely in your browser, with nothing sent anywhere. A URL instead has to be fetched first, which only the server can do (through the same safety checks used across this site's other URL-based tools), before the same scoring-and-selection process runs on the fetched page's text - two different mechanisms arriving at the same kind of summary.
Why word frequency is used as the scoring signal instead of sentence position
A sentence containing several words that recur often throughout the piece is a reasonable proxy for a central idea, since repeated terms tend to mark what the piece is actually about - this is a lightweight, fast heuristic rather than genuine language understanding, which is exactly why it occasionally misses a sentence that states something important in unusual wording that doesn't repeat elsewhere.
Why the selected sentences are put back in their original order rather than ranked by score
Reordering bullets by how high they scored would scatter related ideas and break the narrative flow of the original piece - restoring the original sentence order after selection keeps the bullets reading coherently as a condensed version of the source, rather than as a jumbled list of disconnected high-scoring fragments.