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At the same time as the Steam Client updates itself merely at startup like it appears, it is of no use to have the Steam Client download exposed on top of the download page while a new patch was downloaded or applied. I think we may perhaps furthermore "Play" with the Client and recommence somewhat that can't be resumed by the side of this place. Therefore don't have the Client come into view inside the download list following patches. I am presently utilizing the Beta Update.
AS far as I know this option comes by default without any reason even if you still on the Steam client. If you are using the older version or the beta version of Steam client then the Update or download list normally and all time shows the option for you to download the full version or latest version of Steam client for your computer. Furthermore if the Steam is fails to apply that patch your existing Beta update Steam then also the list will end up with continually showing the option of download.
Every time when you download a patch or any kind of update and apply it to your steam then you will obviously get an option to download steam or may be other version of steam. As you said you are on Beta update. Therefore the steam client might be reminding you that there is another version or updated client there you don’t need to apply patches. However you can remove that option and clear the list anyhow if you don’t want see any stuff of the list.
The Steam client download option showing on the download list or the update list is a fixed option which cannot be closed or removed or deleted from the list. The list will automatically going to update when you connect to the internet. If you have enabled option like automatically download and install patches or something similar then the steam client will do it without your notice. If it doesn’t get any update or patches then as an alternative option. It will show you the download option of new client.
I'm not definite I concur with you for the reason that it's good to be capable to observe while the Steam Client is updating. It doesn't simply update by the side of start-up and can occasionally bring up to date if a patch is launched whilst your Steam Client is previously running. Possibly a superior resolution would be to create an exception intended for the Steam Client list-item with the intention that it doesn't exhibit the Play button similar to additional items inside the Downloads list.
I didn’t see any kind of download option inside the list in my case. It seems to be that the option is only showing inside the beta update list. It would be better for you if you get the prior version steam and see whether there is same thing or not. If there is also same then I can say that you can‘t modify it anyhow. Have you gone through the all setting available for steam for customization? If there any option to disable the refreshing of the list, then do it so that you can manually search for patches when you want.
AS someone said you need to add exception for so that the Steam client list does not show any kind of option like download or install. The patches only contain APPLY option (As I think) so if you add exception to other options then it will no longer show in your list. However I don’t know how to add exception but I hope that there could be way. If you still didn’t get any solution then I urge you to post your issue on Steam community site with the screenshot. Someone will help you there, if there is any solution.