In mysql it is very simple to make use of the userfile. Therefore you can create multiple users on different databases and different priveliges. I know with web connections, it is better to use one user for all connection and queries.
But I delevope a client server application.
What is the best way to do. We using INNODB for recordlocking.

if its a client server application then I'd strongly reccommend one user one userid. so if you have 50 users, thats 50 userids

But does mysql knows who locking the record?

why would MySQL need to now who was locking the record?
if your application needs to know who locked the record, then your appliciation needs to provide that to the db. whether you do that as part of an audit record thats up to you

as to how you identify the user to the db, you do that as part of the connection, through what ever mechanism you are using.... the front end has to estgablish a connection to the db..... normally thats where you would sepcify the userid and password

So conclusion:
- use one database user to connect from client to server on several pc's
- create an own user file with its own priveliges.
- use pooling for several connections
- One client lockes a record and onother client will getting a message by trying to edit it.