(In SHA-512,) what's the difference between message-length field and message length?
The message-length field is a 128-bit bitstring, when the message length is a (non-negative) integer $\ell$ in range $[0\ldots2^{128})$.
The integer $\ell$ is the size of the message in bits, thus 3967 for a 3967-bit message. That would be 31736 for a 3967-byte (that is octet in the context) message. An integer has no intrinsic representation or base.
The message-length field is 128 bits, which value is derived from the integer $\ell$ using big-endian unsigned binary convention. It's first $128-\lceil\log_2(\ell+1)\rceil$ bits first are $0$, which follows from said convention.
The message-length field is used to indicate the actual length of the message, usually in binary form.
I have reservation against this statement: "indicate" would suggest that something or someone would use the length field to find the length $\ell$ of the message, but that's not the case. $\ell$ is encoded in the message length field, but nothing decodes it for the purpose of determining the length of the message. That's not to say the message length field is useless: it's a security feature† towards preventing collisions between messages of different length.
Also "usually" would indicate there are exceptions to the use of binary. That's not the case in SHA-512 or any hash I know. The only variations I can think of (in other hashes) are on endianness, or perhaps counting the length in bytes rather than bits.
† There's another feature with a similar goal in SHA-512: the first of "the padding bits" mentioned in the question, which is set to $1$. Since there is also the length field, from a security standpoint, we could do without that $1$ bit. Security would not be harmed, code woudl be simplified, and that would make computation of messages of exactly 120 bytes nearly twice as fast. But that's not how SHA-512 happens to have been specified.
