difference between dot product and inner product

In my experience, the dot product refers to the product $\sum a_ib_i$ for two vectors $a,b\in \Bbb R^n$, and that "inner product" refers to a more general class of things. (I should also note that the real dot product is extended to a complex dot product using the complex conjugate: $\sum a_i\overline{b}_i)$.

The definition of "inner product" that I'm used to is a type of biadditive form from $V\times V\to F$ where $V$ is an $F$ vector space.

In the context of $\Bbb R$ vector spaces, the biadditive form is usually taken to be symmetric and $\Bbb R$ linear in both coordinates, and in the context of $\Bbb C$ vector spaces, it is taken to be Hermetian-symmetric (that is, reversing the order of the product results in the complex conjugate) and $\Bbb C$ linear in the first coordinate.

Inner products in general can be defined even on infinite dimensional vector spaces. The integral example is a good example of that.

The real dot product is just a special case of an inner product. In fact it's even positive definite, but general inner products need not be so. The modified dot product for complex spaces also has this positive definite property, and has the Hermitian-symmetric I mentioned above.

Inner products are generalized by linear forms. I think I've seen some authors use "inner product" to apply to these as well, but a lot of the time I know authors stick to $\Bbb R$ and $\Bbb C$ and require positive definiteness as an axiom. General bilinear forms allow for indefinite forms and even degenerate vectors (ones with "length zero"). The naive version of dot product $\sum a_ib_i$ still works over any field $\Bbb F$. Another thing to keep in mind is that in a lot of fields the notion of "positive definite" doesn't make any sense, so that may disappear.

A dot product is a very specific inner product that works on $\Bbb{R}^n$ (or more generally $\Bbb{F}^n$, where $\Bbb{F}$ is a field) and refers to the inner product given by

$$(v_1, ..., v_n) \cdot (u_1, ..., u_n) = v_1 u_1 + ... + v_n u_n$$

More generally, an inner product is a function that takes in two vectors and gives a complex number, subject to some conditions.

In my experience, inner product is defined on vector spaces over a field $\mathbb K$ (finite or infinite dimensional). Dot product refers specifically to the product of vectors in $\mathbb R^n$, however.