Buuck Family Questions

Is 'Fredrick Buck' the 'Ernst Friedrich Buuck'?

A Fredrick Buck of Adams county, Indiana, was awarded a Land Patent:

Frederick Buck awarded land patent

Frederick Buck according to the provisions of the Act of Congress of the 24^th of April, 1820, entitled "An Act making provision for the sale of Public Land," for

— Award of land patent to Frederick Buck of Adams county
Indiana

It was awarded on 5 August 1837:

Date of award

fifth day of August in the Year of our Lord one thousand eight hunder and thirty seven…

— Award of land patent to Frederick Buck of Adams county
Indiana

The legal description of tract of land (in Public Land Survey System) is to land in Township 28 North, of Range 13 East, Section 10:

The Land Patent’s legal description of the tract of land:

the South East quarter of Section ten and the West half of the South West quarter of Section eleven in Township twenty 28 North of Range thirteen East in the District of lands subject to sale at Fort Wayne, Indiana, containing two hundred and forty acres.

— Award of land patent to Frederick Buck of Adams county
Indiana

Question: Is this Ernst Friedrich Buuck of Preble township, Adams county, Indiana?

Is This the 1836 Arrival of Friedrich Buuck (born abt. 1800) Family?

Passenger Arrival List on Ancstry.com of Ship Birg Sophie from Bremen, [1?]st arrived August 1836.

Buck family members for: Propsective Arrival of Friedrich Buuck (born abt. 1800) with this family from Bremen
Line No.	Name	Years (age)	Months(age)	Occupation	The Community to which they [generally?] belong	Died on the Voyage
25	Frederick Buck	35	9	Farmer	Wind[heim?]	Michigan
26	Margaret Kleinschmidt	35	-		D[for ditto?]	D[for ditto?]
27	Charolotte Buck	15	6	D[for ditto?]		D[for ditto?]
28	Sophie Buck	12	6	D[for ditto?]		D[for ditto?]
29	Louise Buck	5	-	D[for ditto?]		D[for ditto?]
30	Diedrich Buck	2	6	D[for ditto?]		D[for ditto?]

The children of Ernst Friedrich Conrad Buuck and ?? Kleinschmidt documented in the U.S. Census are:

Name	Birth Date	Place of birth
25	Frederick Buck	35
9		Farmer
Wind[heim?]		Michigan
26	Margaret Kleinschmidt	35
-
D[for ditto?]		D[for ditto?]
27	Charolotte Buck	15
6		D[for ditto?]
		D[for ditto?]
28	Sophie Buck	12
6		D[for ditto?]
		D[for ditto?]
29	Louise Buck	5
-		D[for ditto?]
		D[for ditto?]
30	Diedrich Buck	2
6		D[for ditto?]
		D[for ditto?]

TODO: Find the births of the children in the Windheim church records on Archion.

Transliteration: Baptism Entry No. 128
[Buch?] Nr	Wohn Ort und Haus Num.	männlich	weiblich	Monat Tag Stunde der Geburt	Monat Tag Stunde der Taufe	Vor. und Zu _ Name des Vaters u[nd] der Mutter, [St]and, Gewerbe `_` Alter	Wie lange die Eltern in der [Ehe?]	TaufZeugen Nach[?ischt] des Kind geboren ist geboren ist	Geschlechts und Familien Name des getauft Kindes	der [??] Sohn	die [???] Tochter	[Evidently: Total No. of Children]	[uncertain]	Bemerkungen Todes Tag und Jahr
129	Windheim 109	68	-	[vierten?] 4ten Novbr. Morg. 7 Uhr	10 Novbr	Ernst Diedrich Buuck [?] 29 Christine Wilhelmine Dörmanns 22	28 Novbr 1830	18 Januar 1831 Conrad Dörmann 78. Friedrich Buuck 47	Ernst Friedrich Conrad Buuck	1	1	2	-	[??sch ??ttog].

I believe—though I’m not certain, since the header does not denote this—that the "fourth" written above the date and time of birth may indicate the birth order of the child.

The translation follows below:

Translation: Baptism Entry No. 128
[??] No.	Place of Residence and House Num.	male	female	Month Day Hour of Birth	Month Day Hour of Baptism	Given and Last Name of the Father and the Mother, Occupation, Type of work `_` Age	How long the Parents married?	Baptism Sponsers[godparents] [uncertain] of the child born	Sex und Family Name of the baptized Child	the [number of?] son[s]	the [number of?] Daughters	[Evidently: Total No. children]	[uncertain]	Remarks Day of Death and Year
[??] 129	Windheim 109	68	-	[fourth] 4th Nov. 7 a.m.	10 Nov.	Ernst Diedrich Buuck [?] 29 Christine Wilhelmine Dörmanns 22	28 Nov. 1830	18 Janary 1831 Conrad Dörmann 78. Friedrich Buuck 47	Ernst Friedrich Conrad Buuck	1	1	2	-	[uncertain].