What exactly Membership Service Provider (MSP) is in Hyperledger Fabric?

OK, so I was able, hopefully, to answer my question. I will answer it in the form of points.

• The name "MSP" is, in my opinion, a poorly chosen name. The word "service" implies having a working process/program/server in place, like Web Service, Rest Service etc. The word "Provider" also implies that it provides the membership service and finally I imagine that this is some kind of an OAuth or LDAP Server or something alike, it is not. It a bunch of directories with digital certificates. Probably a better name would be Members Certificates Directory (MCD) or Members Certificates Registry (MCR).
• After realizing that these are directories from the documentation

To set up a local MSP (for either a peer or an orderer), the administrator should create a folder (e.g. $MY_PATH/mspconfig) that contains six subfolders and a file

Also set up implies starting and configure a process/server or something alike.

• So I entered the peer container of the Dev Servers created for Hyperledger Composer development and searched for a $FABRIC_CFG_PATH

• the /msp DIRECTORY is the MSP directory and it contains directories explained in the docs And it is configured as required by the documents:

The path to the mspconfig folder is expected to be relative to FABRIC_CFG_PATH and is provided as the value of parameter mspConfigPath for the peer, and LocalMSPDir for the orderer

I hope this clears uncertainty for others as it did for me and the docs of fabric updated for more clarity.

Hyperledger fabric 2.2. documentation explains it as per the accepted answer.

https://hyperledger-fabric.readthedocs.io/en/release-2.2/membership/membership.html

What is MSP

Despite its name, the Membership Service Provider does not actually provide anything. Rather, the implementation of the MSP requirement is a set of folders that are added to the configuration of the network... Whereas Certificate Authorities generate the certificates that represent identities, the MSP contains a list of permissioned identities.

Two types of MSP :

Local MSPs are represented as a folder structure on the file system

Channel MSPs are described in a channel configuration.