Short answer for the ArcGIS Administrator - this is what "users" use to authorize software (single-use) and/or point to a (concurrent-use/floating) license manager. If you have concurrent license instead of single use licenses, your License Server Administrator setups up the license manager and provide you with the info to connect to it.
longer answer....
ArcGIS License Server Administrator - the "concurrent" (floating) license manager is a separate software install and is authorized using concurrent/floating licenses. Unless things have changed (since 2016), those are no longer available for purchase unless you already have some and are grandfathered in. This format goes back to the ArcInfo Workstation days and is/was a way that you can share a set number of license with many more users. Since many users use it sporadically, this is a great model.
The license manager started supporting concurrent use for Pro a year or so ago. This is managed thru the MyEsri site, as are the ArcGIS (ArcMap/Catalog) concurrent licenses. On this site you request a provision file and then authorize using the ArcGIS License Server Administrator software.
If you are a user and not the administrator of your account, you typically only have to worry about the first option, with the info (server/IP) your agencies license administrator provides you, if appropriate.
Clear as mud?
...of course, if you have named users only, that is different still.
edit: sorry, got caught up on explaining the difference, that i didn't answer any other part of the question.
The authorization codes for 10.5 were different than than 10.1-10.4. Usually yyou should always update your LM to the latest version, assuming your have the authorizations that are valid for it. If you have 10.5 authorization codes, you should be on LM 10.5...and should be able to upgrade to 10.5.1. I just downloaded and plan to do that tonight. We have users that access our 10.5 LM that have any software from 10.1 - 10.5.0 currently installed. I have converted some of our Pro from named users to concurrent, so if I upgrade, once 2.0 is released (very soon) we will be ready. Typically they release the LM at the same time as the software....this is a first, but nice to get prepped. Enough rantng.