I am no expert in the history of Mathematics but I feel I must say the following since no one else has. Of course Euler did not think that $1 - 1 + 1 - 1 + 1 - 1 +\cdots=\frac12$ in that way.

Let's agree that we know what a real series is, essentially any "vector " in $\mathbb{R}^{\mathbb{N}}$ with a $\Sigma$ in front (or $+$ signs) to tell us that we are not looking at it as a sequence. The sequences form a real vector space. There is a subspace which we would call the convergent sequences and we allow ourselves to say things like $\sum 2^{-n}=1$ although we might claim that really we mean that there is a linear transformation $S$ whose domain is the convergent series with range $\mathbb{R}.$ We sometimes symbolically enlarge the range in a familiar way to include $\pm \infty$ to discuss some kinds of "divergence."

There is a venerable subject of divergent series and "summation" methods for them. Such a method is a linear transformation $T$ which agrees with $S$ on the "convergent" series and perhaps a few other axioms enjoyed by $S$ (adding a few extra terms changes the sum in the obvious way.) Any method satisfying those requirements and sending $1-1+1-1+1...$ to a real $r$ would have to send it to one with $r=1-(1-1+1-1+\cdots)=1-r$ and hence to $\frac{1}{2}$

Maybe most mathematicians have no use for the distinction between a convergent series and the number it converges to. I would certainly advise my students to avoid summing divergent series until they had a firm grasp on the usual practices.

After that I find it a very interesting topic but not a useful one for me. I think this is not practiced much these days , perhaps with good reason. (I guess it is called regularization when it is Quantum Physicists making the mathematical formalism conform to measurements.)

It is a valid subject and Euler understood it well if not in our exact formal approach.

I am no expert in the history of Mathematics but I feel I must say the following since no one else has. Of course Euler ( who just had his $17\cdot 18$) did not think that $1 - 1 + 1 - 1 + 1 - 1 +\cdots=\frac12$ in that way.

Let's agree that we know what a complex series is (stick to $\mathbb{R}$ if you wish): essentially it is any "vector " in $\mathbb{C}^{\mathbb{N}}$ with a $\Sigma$ in front (or $+$ signs) to tell us that we are not looking at it as a sequence. The series form a complex vector space. There is a subspace which we would call the "usual" convergent series and we allow ourselves to say things like $\sum 2^{-n}=1$ although we might claim that really we mean that there is a linear transformation $S$ whose domain is the convergent series with range $\mathbb{C}.$ We sometimes symbolically enlarge the range in a familiar way to include $\pm \infty$ to discuss some kinds of "divergence." And of course we also care about the assertion $\sum z^{-n}=\frac{1}{1-z}$ which is unproblematic in the closed unit disk with the exception of two interesting points which deserve further thought. Integrating term by term (a bold move?) suggests that $-\sum \frac{z^{n+1}}{n+1}=\ln{(1-z)}.$ Whatever our qualms about the first series at $z=-1$, the second seems true and it looks like magic. The convergence is slow although there are ways to accelerate it. These methods applied to $\sum z^n$ at $z=-1$ lead to a not unexpected result.

There is a venerable subject of divergent series and "summation" methods for them. Such a method is a linear transformation $T$ which agrees with $S$ on the "convergent" series and perhaps satisfy a few other axioms enjoyed by $S$ (adding a few extra terms changes the sum in the obvious way.) Any method satisfying those requirements and sending $1-1+1-1+1...$ to a real $r$ would have to send it to one with $r=1-(1-1+1-1+\cdots)=1-r$ and hence to $\frac{1}{2}.$

Maybe most mathematicians have no need for the distinction between a convergent series and the number it converges to. I would certainly advise my students to avoid summing divergent series until they had a firm grasp on the usual practices. The perspective of "summing divergent sequences" feels a bit quaint these days but it is a valid subject. The amazing and brilliant things Euler and others did is the reason we now may choose to safely explore have well tamed domains such as analytic continuation of functions of one or more complex variables or regularization as practiced by Quantum Physicists making the mathematical formalism conform to measurements.

I'm not sure how all this connects to the actual question above but I'm not convinced that it is unrelated.