Is a modular home considered a mobile home when a woodstove is added for insurance purposes?

A modular home is a Pre built stick home, not a mobile unit. If your insurance agent says otherwise , look for a new agent.
Wood burning stoves have to have wall clearance and should have a protective flooring under it (asbestos board) or something similar. Also double wall insulated pipe where the pipe goes through the structure.

If it was parked and had axles ,it is a mobile home. If it was brought out and set permanently ,it would be a modular home. Just because you put a fireplace in does not change the status of the dwelling. but it does change your insurance ratings!! There are some site built homes out there now,so be sure for insurance reasons exactly what you have. I looked at a property that was beautiful,but something wasn't quite right with it. I looked under the home and behold,it was a modular that was totally bricked up!! Beautiful job but what would the insurance call it? Depends on what your state laws/guidelines say. You will have to check the codes in your area to be sure. I hope this helps!!

Whether the home has a wood stove or not has nothing to do with the classification between "modular" and "mobile".

A "mobile" home is designed to be able to move it again at a later date... typically, the steel frame is left intact underneath the home. It CANNOT be removed, without structurally altering the home.

A "modular" home is built so that the steel frame is REMOVED once the home is set up. The "wood frame" is built differently from a 'mobile' home... and would be more like you would find in a 'sitebuilt' home. The structure is self supporting and does not need the steel to remain.

For insurance purposes... you can make a "mobile" home a permanent fixture... this involves removing the wheels and axles (normally done anyway), and perimeter blocking the home. This satisfies most insurance companies definitions of a "permanent" home, as opposed to a "mobile" home.

Mobile homes can have fireplaces and woodstoves, and still be "mobile" - and if anything, will INCREASE the insurance cost of the home. Making the mobile home a permanent fixture though, will result in a decrease in the insurance premiums. And don't forget to check the manufacturer's warranty... some types of stoves can void your homeowner's warranty (Mobile OR Modular).

Check with your local insuarance agent for exact requirements to satisfy them, and check your homeowner's warranty.

Have Fun

A woodstove would not change the description of your property. Whatever it says on the bill of sale, that is what it remains. However, I am a little confused by your question, why would your insurance require a woodstove? It seems like it would be the opposite.