There is confusion about the Elliptic curve terminology in this question. Let deal some of them;
Elliptic Curve
Algebraically an elliptic curve is
$$E(\mathbb{K}) := \{ (x, y) \in \mathbb{K}^2 \mid y^2+a_1xy+a_3y = x^3+a_2x^2+a_4x+a_6\} \cup \{\mathcal O\}$$
$\{\mathcal O\}$ is the point at infinity added as extra that has no representation in the geometric shape of the curve.
The points are the $(x,y)$ tuple that satisfies the curve equation so they are not integers!
Point addition
The point addition has a very well geometric meaning. In the below picture $P,Q,R$ represents the points on the curve and $\{\mathcal O\}$ is represented as $0$
and we extract the arithmertic equations from this ( tangent chord rule). For detail of the extraction look at Chapter 2 of Washington's book.
The points of a curve form an Abelian group under the point addition operator with the identity element $\{\mathcal O\}$.
Scalar multiplication
When we add a point $P$ to itself we say doubling some person write as $2P$, however, the common and better way to write it is $[2]P$. So $[2]P = P + P$.
Similarly, we can talk about adding three times, four times, or $t$ times.
$$[t]P : = \underbrace{P + P + \cdots + P}_{t-times}$$
This is what we call the scalar multiplication ( actually a Z-Module for Abelian groups)
Generator
A generator of a cyclic group is an element $G$ such that when $G$ added itself again and again it will generate all elements of the group (Sorry for the group theorist, the capital letters colliding here - an element $g$ of a group $G$ is generator if $\langle g \rangle = G$).
Order
The order has two usages in ECC
Order of the Elliptic curve $|\#E(\mathbb{K})|$ means the number of elements of the curve
Order of an element.
When the curve has prime order as in Secp256k1 then every element has the same order as the curve order and this implies every element is a generator.
Back to your question
In Secp256k1, the base point
G = (55066263022277343669578718895168534326250603453777594175500187360389116729240,
83121579216557378445487899878180864668798711284981320763518679672151497189239 )
and the order of the basepoint $n$ is
n = 115792089237316195423570985008687907852837564279074904382605163141518161494337
The order means that $[n]G = \mathcal{O}$ and we can uses this to derive the below equation
$$[k]P = [ k \bmod n]P$$
So what you do is with $+2$ is
$$[n-1]G + [2]G = [n-1+2]G = [n+1]G = [1]G = G$$
So what you do is with $+1$ is
$$[n-1]G + [1]G = [n-1]G = [n]G = \mathcal{O}$$
Let's finish with SageMath verification;
#secp256k1
p = Integer("0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F")
a = Integer("0x0000000000000000000000000000000000000000000000000000000000000000")
b = Integer("0x0000000000000000000000000000000000000000000000000000000000000007")
K = GF(p)
E = EllipticCurve(K,[a,b])
print(E)
G = E(Integer("0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798"),
Integer("0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8"))
print("\nG =",G)
n = G.order()
print("\nG's order =",n)
G2 = 2*G
Q = (n-1)*G + 2*G
print("\n[n-1]G+[2]G =",Q)
assert(Q == G)
R = (n-1)*G +G
print("\n[n-1]G+G =",Q)
print(R)
and the output is
Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 115792089237316195423570985008687907853269984665640564039457584007908834671663
G = (55066263022277343669578718895168534326250603453777594175500187360389116729240 : 32670510020758816978083085130507043184471273380659243275938904335757337482424 : 1)
G's order = 115792089237316195423570985008687907852837564279074904382605163141518161494337
[n-1]G+[2]G = (55066263022277343669578718895168534326250603453777594175500187360389116729240 : 32670510020758816978083085130507043184471273380659243275938904335757337482424 : 1)
[n-1]G+G = (55066263022277343669578718895168534326250603453777594175500187360389116729240 : 32670510020758816978083085130507043184471273380659243275938904335757337482424 : 1)
(0 : 1 : 0)
