Can a website use multiple meta keyword tags?

The keywords meta tag doesn't do anything anymore, at least as far as most search engines are concerned. You're trying to solve a problem that doesn't exist. Search engines index by content nowadays, and they do that without your help.

See the Wikipedia article on the meta tag:

Search engines began dropping support for metadata provided by the meta element in 1998, and by the early 2000s, most search engines had veered completely away from reliance on meta elements. In July 2002, AltaVista, one of the last major search engines to still offer support, finally stopped considering them.

Your first example is the correct way, the second would not improve things at all, and might make things worse.

Personally, I disagree with Welbog's answer, primarily on the account that if keywords were no longer relevant, why do things like Google Webmaster Tools cater for them? I suspect that there might have been some confusion with meta description tag, which did go out of favour for some time a while ago, but does appear to be coming back into fashion.

When optimizing individual pages, title, keywords and description tags are all important, as are long tail URL strings and directory names, but don't abuse them. Also, you can make clever use of the author and copyright tags by rewording your company name or title in them again.

It is possible to find good and bad reasons to use/not use any tag. What you will find though is generally the following. Those that suggest a tag is not needed are invariably selling a link program or PPC type thing. Those that use them and take the time to use them inventively without breaking the rules will charge more up front, but, the long term benefits will be huge.