I downloaded a family of Latin Modern font from latex website some time ago. It's licensed under LPPL license, but I still don't understand if that means that I can use it on a public website, somewhat dedicated to profit.
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You can. The LPPL does not restrict the use of the WORK (which is what you are asking about) the license only defines under which conditions the WORK can be distributed and modified.
The word "running" may sound a little strange in the context of fonts, but that is due to the license initially being designed for software programs.
In short, you can use any WORK under this license for whatever you wish including commercial usage.
Note: I'm one of the principal authors of the license, so you can see the above statement as a notice of intent.