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I have been able to track down the website of Robin Barry Beard a genealogist whom I relied on extensively for my previous Mackinder family trees (see #12). His is still very active and has posted his family trees (including an extensive one for Mackinder) on his own website http://what-a-family.co.uk. Access is free but you need to register with a username etc.
Robin Beard now shows James(b1546 in Aubourn) and Alice(about 1545) Mackinder as the parents of Alice(b 1562), Artemas(b 1563) and Robert(b 1575) Mackinder. He shows James' father to be another James Mackinder born about 1525. His only source I can see is "extracted probate records for Jacob Makender" but he doesn't include Jacob Mackinder anywhere in the tree.
I will try and contact Robin Beard more directly to understand his sources.   

Thanks, Machis, for sharing this historic document. It explains the letter I received in 1986, unsolicited and purely on the basis of my surname. That letter referred to two Scottish brothers who each purchased 1000 acres of land in Lincolnshire in the 1700s - clearly this is straight from Sir Halford. Unfortunately the note in his Sir Halford's father's papers must have omitted at least four generations prior to John II & Elinor: ie John I (b 1673?), Joseph(b1621?), William(b1595?) and Artemas(b1575), who were all born in Lincolnshire.
The "Mack-an-Deoir" derivation of our surname is replicated in a "handwritten history" of the surname that I purchased in 1995. Do we think this is based purely the supposition of the Duke of Argyll? My handwritten history claims there are documents referring to a Colin Macindoyr in Fife c.1250. I also note the in the Lincolnshire Archives there is a book "Long Road from Loch Awe. A Mackinder Family History". Does anyone know how I can get hold of a copy, and does it contain more substantial data on the Scottish connection? 

Wow! Thanks so very much wildwitch for sharing your extensive & painstaking research. It seems each piece of the jigsaw adds more intrigue without ever quite closing the circle! I will need to pour through it many times more, together with the conclusions of Machis, before I get close to understanding these early relationships.
Nine years ago, when I was researching the family tree for my late father, I relied for my 16th & 17th century data on a tree placed on an ancestry website by a Robin Beard, who I was never able to make contact with. I was unable to substantiate any of his data, and much now seems inaccurate. Robin had Artemas as 1575-1642, which, if I understand correctly, should be 1562-1624. He had Artemas' father as Jacob (1550-1606), which again seems false. In addition to the sons & daughters named in Artemas' Will, Robin also has recorded Artemas having an eldest son, John (b1591), who had three children, Elizabeth (b1611), Francis (b1611) and Thomas (b1614). Is this correct?
Does anyone know about the Mackinder clocksmiths of Spilsby, Lincolnshire? I have recently acquired a grandfather clock supposedly made by a William Mackinder in the 1840s, which has "Mackinder Spilsby" on the clock face. Initial searches suggest there was a family of three clocksmith Mackinders in Spilsby named Thomas, Townson and William. Hence I am guessing they could be Townson (1777-1836), his son William (b1805) and possibly Townson's uncle Thomas (1723-1779)?