Hi Val, 


My understanding of Web fits with the definition of web page when talking in the context of "web applications" or "web pages".  It does not define whether it is online or not.
http://en.wikipedia.org/wiki/Web_page
A web page (or webpage) is a web document that is suitable for the World Wide Web and the web browser.

This is typically an html page wether the file extension is .htm, .html, .php, .asp. .aspx, .jsp etc.

Whereas Web can also mean the systems of linked web pages on the World Wide Web  on web servers whose domains usually but not always starts with www in the context of Internet as a network.
So "on the web" to me would mean that you need an Internet connection to access it.

In the programming world there are three kinds of apps for phones and tablets.
Native Mobile Apps
These could be considered analogous to "Desktop" on a computer.  It installs on the device and can be used even when internet is not available.  It is developed for a specific platform either IPhone, Android etc. and has access to the files and other hardware on the device. 
http://searchsoftwarequality.techtarget.com/definition/native-application-native-app

Mobile Web Apps
Web sites made specially to be used on a mobile device hosted on the Internet.  Usually work on most mobile platforms.  Can only be used when an internet connection is available.  Cannot access the local files.
http://searchsoftwarequality.techtarget.com/definition/Web-application-Web-app

Hybrid App
A web application that is hosted the device and can access the device's files. Can be used even if an internet connection is not available.  Has access to the local files.
http://searchsoftwarequality.techtarget.com/definition/hybrid-application-hybrid-app


    Sometimes it depends on how you are classifying things.  Whether from a network or non-network perspective or from a web browser non-web browser perspective.

	web browser	non web browser
network	Web (Like SignPuddle), online, Internet, "on the web", Mobile Web App
non network	Web (Like Personal Puddle), offline, Hybrid App	Desktop, Native Mobile App

Jonathan