The second and the third are pretty much equivalent. Indeed, "the period" in XIX century sense is essentially the same as the discrepancy between the branches of a multi-valued function, obtained as an integral of an algebraic function. If you take the Riemann surface associated with this algebraic function (that is, a branched covering of C where it is well-defined), the period becomes an integral of the holomorphic differential $fdz$ associated with this function over a closed loop, that is, an integral of a holomorphic 1-form over a closed cycle.

The modern definition of "periods" is, more or less, "the pairing between holomorphic differential forms and integral homology". However, this notion can (and often is) extended to Hodge structures, and this is related to the first notion you mention.

Define the Teichmuller space of a complex manifold as the space of all complex structures (here the complex structure are understood as endomorphisms of $TM$ satisfying $I^2=-Id$ and the integrability condition) up to isotopies: $Teich= \frac{Comp}{Diff_0}$. Never mind that this space might be horribly non-Hausdorff; the period map is naturally defined on $Teich$, and (if you are lucky) it would help to make sense even of the non-Hausdorff pathologies.

The period map is the map taking $I\in Teich$ to the Hodge structure on $H^*(M,I)$, which is understood as a point in the appropriate flag space. This map is holomorphic, and, if you are lucky, defines a biholomorphism between $Teich$ and the space of Hodge structures ("period space"). This holds, unfortunately, only for complex tori, but weaker versions of this statement are true for K3, hyperkahler manifolds, complex curves and in some other cases. These results are called "global Torelli theorems".

The local Torelli theorems are statements about the local structure of the period map, saying (in most cases) that is it locally an immersion; this is true, for instance, for Calabi-Yau manifolds.

The modern definition of periods is not entirely equivalent to the traditional, because the Hodge structure contains more data than just holomorphic forms; however, if you know the Hodge structure on cohomology, you know the pairing between the holomorphic forms and the integer homology, hence it is an extension of the XIX century notion.

If you restrict yourself to the first cohomology, the Hodge structure is a flag $H^{1,0}(M)\subset H^1(M)$; in this case the XIX century notion of "periods" coincides with the Hodge-theoretic notion.

The second and the third are pretty much equivalent. Indeed, "the period" in XIX century sense is essentially the same as the discrepancy between the branches of a multi-valued function, obtained as an integral of an algebraic function. If you take the Riemann surface associated with this algebraic function (that is, a branched covering of C where it is well-defined), the period becomes an integral of the holomorphic differential $fdz$ associated with this function over a closed loop, that is, an integral of a holomorphic 1-form over a closed cycle.

The modern definition of "periods" is, more or less, "the pairing between holomorphic differential forms and integral homology". However, this notion can (and often is) extended to Hodge structures, and this is related to the first notion you mention.

Define the Teichmuller space of a complex manifold as the space of all complex structures (here the complex structures are understood as endomorphisms of $TM$ satisfying $I^2=-Id$ and the integrability condition) up to isotopies: $Teich= \frac{Comp}{Diff_0}$. Never mind that this space might be horribly non-Hausdorff; the period map is naturally defined on $Teich$, and (if you are lucky) it would help to make sense even of the non-Hausdorff pathologies.

The period map is the map taking $I\in Teich$ to the Hodge structure on $H^*(M,I)$, which is understood as a point in the appropriate flag space. This map is holomorphic, and, if you are lucky, defines a biholomorphism between $Teich$ and the space of Hodge structures ("period space"). This holds, unfortunately, only for complex tori, but weaker versions of this statement are true for K3, hyperkahler manifolds, complex curves and in some other cases. These results are called "global Torelli theorems".

The local Torelli theorems are statements about the local structure of the period map, saying (in most cases) that is it locally an immersion; this is true, for instance, for Calabi-Yau manifolds.

The modern definition of periods is not entirely equivalent to the traditional, because the Hodge structure contains more data than just holomorphic forms; however, if you know the Hodge structure on cohomology, you know the pairing between the holomorphic forms and the integer homology, hence it is an extension of the XIX century notion.

If you restrict yourself to the first cohomology, the Hodge structure is a flag $H^{1,0}(M)\subset H^1(M)$; in this case the XIX century notion of "periods" coincides with the Hodge-theoretic notion.