So the best solution for nowadays is to use just char, short, int and long long (and make strong assumptions that these are exactly 8, 16, 32 and 64 bits wide respectively), never use long or long double. Never use (u)intNN_t. Then you are good.
Those caveats of the past (but int might be 16 or 36 bits), are exactly that. An artifact of the past. A historical curiosity. Not relevant for today or the future. No, I don't believe for a second that any future platform will change their size.
Platforms also still disagree on the signedness of char, so when an 8 bit numeric type (as opposed to an ascii character type) is needed, one should always explicitly specify signed char or unsigned char, both of which are separate types from char.
Further things of note: platforms also have agreed on little endian (so called "network byte order" is dead and should never be used in new protocols, because it forces everyone to convert) and on IEEE memory representation of float and double. Contrary to popular belief the main floating point operations (+,-,*,/,==,<,>,<=,>=) are also precisely defined and always behave exactly the same (leaving out strange edge cases such as denormals). And yes, of course platforms have very long agreed on twos-complement for negative integers. This even made it into the standard at some point, I believe. Same happened with the memory layout of a vector<>, which in the past wasn't standardized, but because everyone of course did the obvious (and made it the same as a normal C array), it was added to the standard later.
What I am saying, what the C++ standard guarantees isn't everything. There are much more guarantees modern C++ code can (and should) rely on.