Open source isn't just open source. There are lots and lots of open source licenses with different conditions.
So-called permissive OS licenses like BSD or MIT allow you to do almost anything you want with the covered work, even relicense it under a completely different license, as long as you include a copyright notice.
But there are also share-alike OS licenses like the GPL which only allow derivates to be released under the same license. When you use a GPL library, any software which links to it statically must also be licensed under the GPL. Because this is very inconvenient for libraries, there is a special version of the GPL, the LGPL. LGPL stands for "Lesser GPL" or "Library GPL". It allows to place software which uses the library under any other license as long as you don't make change to the library itself. And even when you do, you just have to distribute the changes to the library under LGPL, not your own application.
And then there are lots of other licenses with their own conditions. So what you should do now is find out which license is used by each of the libraries and then find out what they allow.
I believe you've stated the differences between the Mozilla Public License and the GNU Lesser General Public License accurately, and either may suit your needs just fine, but you are skipping over the most important difference between the two licenses:
Who can make new versions?
Both the MPL (section 10) and the LGPL (section 14) include in their license grants the right to substitute the current version with a latter version, and there are no actual limitations as to what can go into those licenses. While it's highly unlikely that either the Mozilla Foundation or the Free Software Foundation will do something as crazy as, say, institute a clause that says "all contributions to this software become our property", it's not beyond the realm of possibility that one of the organizations will release a new license version that you don't like.
Which brings up another point about using a "Modified LGPL".
A modified license is not the same license!
While you have fairly amazing ability to specify your own licensing terms, and could in essence say "you can distribute this as per the GPL, but you need to put my name in your credits and pay me 1% of any revenue you generate", any time you do so you are creating a new license based on someone else's work. This means that you're NOT using the MPL or the LGPL, you're using a new "mucaho license".
What that means is that you probably won't get any help from your original license's author if you need to defend your interpretation of the license inside of a courtroom, and it's entirely possible that they might file suit to say that THEIR version should apply and not yours.
Of course, both of these are minor points. Even "license popularity" doesn't matter unless you expect your code to be directly incorporated into larger projects.
Personally, I think the MPL is a better choice if you like proprietary compatibility, or if the choice is between the actual MPL and a different license you have to manually edit based on the LGPL. Unless you have a reason not to use the MPL, go with something backed by a foundation instead of one that might leave you in a courtroom without any aid whatsoever.
Best Answer
It should be sufficient to just read the MPL.
So the MPL isn't viral in the sense that the GPL is. MPL software can't be re-licensed under your own incompatible license, but you can distribute it with any other code under any other license as long as the other license doesn't conflict with the MPL (as the GPL does), and as long as you still distribute the MPL source.
Also, if you're not distributing the software to anyone, then the discussion is moot, as you're not required to distribute source to anyone to whom you don't distribute binaries.