In today's English, you sit yourself at the table. The example from your second link is outdated (it is from the early 1800s).
In the two sentences you ask about,
My goodness what do we have here? You look as though you have been traveling for a very long time. Please ---- sit yourself at the table. I will bring you something to wash your hands.
When you receive company, show great kindness to all you receive; invite them to make good cheer; let them be well served...When the day appears, do not set yourself at the table to eat alone: nothing is more unpolite. Place your guests near you in a neat apartment [apartment here means "room"].
there is NO difference in meaning between
sit yourself at the table
set yourself at the table.
In regard to these particular sentences, both sit and set are
1. transitive verbs
2. used in the imperative mood
2. meaning to cause to sit or to seat
3. with a reflexive pronoun as the direct object
In sum, the meaning in both sentences is Cause yourself to sit down at the table or Seat yourself at the table.
it is important to note
1) Set and sit both have a long history in English (we're talking since the 700s or 800s).
2) There is a technical difference between set and sit.
3) This difference has been mistaken and/or ignored for about 700 years. ("Confusion between set and sit arose as early as the beginning of the 14th century"--Oxford English Dictionary.)
4) The second passage you link to is from a book published in 1817. Its usage of set is outdated. In contemporary English, only some dialects maintain this usage of set.
So the "mainstream" of native English speakers would not use set to refer to people being in or placing themselves or others in a sitting position. So, if you want to sound like most modern Native English Speakers, use sit at the table.
Examples of older uses (From OED):
1719 D. Defoe Life Robinson Crusoe (Globe) 15 We..set us down to fish.
1845 Dickens Chimes i. 30 You must always go and be a settin on our steps, must you!
1848 Thackeray Vanity Fair lv. 490 I'm thinkin' if I set here until I'm paid my wages, I shall set a precious long time, Mrs. Raggles; and set I will, too.