I have emailed Lincoln archives to see whether the elusive 16th century parish register for Aubourn is in their possession somewhere, but am yet to hear. All of the parish registers held at Lincoln archives though are allegedly recorded (and mostly scanned) on their Lincs to the Past website and can be found by entering Aubourn par1 into the search bar -
https://www.lincstothepast.com/SearchResults.aspx - the earliest I can find on their website is 1749.
I have also not found original parish register images (not transcripts!) for Aubourn on Find my Past before 1749. You can see clearly on Find my past and Family search (unless I am searching wrongly) that there are early BTS transcripts for Aubourn only for the years 1562/3, 1565/6 and 1569/70 and then they continue annually after 1587/8. Parish registers were not kept in England until 1538 and only very few Lincolnshire parishes have surviving parish register books going back this far. Early BTS records are often incomplete and were written on sheets of parchment and submitted by each parish on an annual basis from around 1562 onwards, but many parishes did not start completing these until the 1590s. We are therefore lucky with our Aubourn records! Before 1587 therefore the only transcribed registers are for those 3 years, which would be surprising if there were a surviving parish register book in the public domain for this time period for Aubourn. The BTS for 1569/70 shows no evidence at all of a baptism for any Artemas Mackinder (this register ran from Michaelmas). I have original copies of them all.
The earliest BTS for Aubourn is 1562/3 and shows the two Mackinder baptisms (Artemas and Alice) that have been transcribed on Find my Past and Family Search. All 3 of the early Aubourn BTS records (including 1562/3) ran from Michaelmas to Michaelmas (as the original 1562 record states! 'all the names & surnames of such persons as were married, christened and buried within ye town of Auborn from the feast of St Michael the Archangel which was in the year of our lord god 1562 unto the same feast 1563'). This means that both Alice and Artemas were baptised in 1563 by our current calendar within 3 months of each other. Alice on 21.2. and a few entries later Artemas on 20.5. This means they are unlikely to be full siblings, although neither entry records the name of any parent. Unfortunately none of the marriages and burials have been transcribed for these 3 registers, but the original records in regards to this show that a Robert Macater was church warden in 1562/3 and a Robert Macat'r was buried on 25.4.1565/6. All of these original BTS records can be obtained and probably ordered in digital form from Lincoln archives, although I am not sure to what degree records can currently be ordered due to COVID and thus working regulations.
I have therefore not seen any baptism for Jacob for 1550. As previously mentioned the earliest records for any Mackinder I can find for Aubourn are for a William Mackant's, who died in 1555. The original inventory can be seen here:
https://www.lincstothepast.com/Inventory---Mackant-r--William--1555-1556--------/1384676.record?pt=SI first seem to find this William Mackinder as one of the men who drew up Thomas Chamberlayn's inventory in Aubourn on 8.5.1541:
https://www.lincstothepast.com/Inventory---Chamberlayn--Thomas--8-May-1541--------/1076112.record?pt=SAs previously mentioned by the 1560s and 1570s there appear to have been two adult Mackinder men in Aubourn: Robert, who probably died in 1565/6 and Thomas (who may have died in 1608). There are though two other BTS entries of interest for Aubourn: 1589/90 Jane Macander was buried 27th July. Sibell Makender widow was buried the 17th April 1595. Sibell was clearly somebody's widow and could also even potentially have been Artemas's mother, but we will probably never know.