The way it is meant to work is that meteor installs its own version of node, and this should not clash with what you have already installed. The meteor dev bundle includes the packages that should be needed, in a node_modules directory.
For me this is:
<users home directory>/.meteor/packages/meteor-tool/.1.5.0.q9pe82.mj0xhzd7vi++os.osx.x86_64+web.browser+web.cordova/mt-os.osx.x86_64/dev_bundle/lib/node_modules
When Meteor's node executes it should find the packages it needs there first, but if it doesn't, the standard node search path is used, which involves a search up to the root node_modules and your home directory node_modules. As the meteor tool runs from inside your home directory it may always end up looking at ~/node_modules.
So the big question is why didn't it find the version of sqlite3 in the meteor-tool folder?
My guess is that you have some node environment variables set that are overriding things. Possibly having a different version of node in your PATH might also do it.
Check for NODE_PATH especially.
Ultimately, I think if you remove the environment variable when running meteor it should be OK. This means that they can co-exist on the same machine so long as you use a different console with the environment tweaked.