Dotnet Core vs DotNetStandard
The two are not "vs" each other. Rather, .NET Core "contains" an implementation of the .NET Standard Library (as well as extra stuff that is not in .NET Standard). Here it is as a Venn diagram.

...how is it possible that e.g. google API sheet support dotnetcore with dotnetstandard v1.3?
It is possible because .NET Core 1.0 supports version 1.3 of the .NET Standard Library.
In the following table, netcoreapp
is .NET Core, net
is the .NET Framework, and netstandard
is the .NET Standard Library. The .NET Standard Library, as you wrote, is a subset of functionality.
Important: Each platform advertises the highest version of the .NET Standard Library that it supports.
netstandard 1.0 1.1 1.2 1.3 1.4 1.5 1.6 2.0
netcoreapp → → → → → → 1.0 2.0
net → 4.5 4.5.1 4.6 4.6.1 4.6.2 vNext 4.6.1
Here are some examples to check your understanding.
- .NET Core 1.0 supports up to .NET Standard Library 1.6
- .NET Framework 4.6.1 supports up to .NET Standard Library 1.4
.NET Standarded Library 1.3 is supported by...
- .NET Core 1.0 and 2.0
- .NET Framework 4.6, 4.6.1, 4.6.2, and vNext
...what do i have to install to allow applications using dotnetstandard 1.3 run under dotnetcore?
You have to install .NET Core.