Why is an ionic bond a chemical and not a physical bond?
Short version:

We don't call bonds "physical", there are chemical bonds and other types of interactions between particles. The chemical bonds are classified this way because they make up molecules, salts, polymers and such, which are the materials chemists are interested in studying, and not particularly because of their sub-atomic/electrostatic characteristcs. 



Your question relates to two frequent sources of confusion: 


How do we define a chemical bond? What makes a bond between two atoms ionic or covalent?
Where does chemistry (hence chemical interactions) ends, and consequentially where does physics (and hence physical interactions) begins?


Both questions have been pretty much answered here on Chemistry SE and also on Physics SE. I'll give some links to related questions and also a shrot summary.

1. How do we define a chemical bond? What makes a bond between two atoms ionic or covalent?

Your question didn't go much into covalent bonds, but anyone trying to understand anything past the surfaces of chemical bonds should know that the definition of a bond as covalent or ionic is simply an easier way for us to classify things, because when the bonds are categorized in groups we can say that one group behaves in a way, and the other behaves in a different way. The truth is, all bonds have some ionic and some covalent character to them. 

Good answers here on Chemistry SE about that can be found here: 


Can 100% covalent bonds exist?
What happens if the electronegativity difference is exactly 2.1?


2. Where does chemistry (hence chemical interactions) ends, and consequentially where does physics (and hence physical interactions) begins?

For a Physics pont of view, see: What distinguishes between physics and chemistry?

For a Chemistry point of view, see: Differences between chemical physics and physical chemistry?

Basically, it's clear that both sciences have a lot of overlapping, and perhaps the main difference is the approach taken and how the results will be used. While chemists usually focuses on understanding the structure and interactions between substances and materials, in order to be able to improve them or develop new ones, physicists are mostly interested in understanding the exact laws of nature and representing them with mathematical equations that can be used to explain and predict phenomena. This definitions do not seem so different, and that's not supposed to be wrong: We do have the Physical Chemistry and Chemical Physics fields.

Ok, but Why is an ionic bond a chemical and not a physical bond?

Two important things were mentioned by @Mithoron in the comments. The term "Physical bond" is not frequently used by chemists, (or anyone I suppose), pretty much because all interactions in whole universe are physical. This single quote by @Mithoron is related to the two points I highlighted above. Both "types" bonds "follow" the same laws, and there is no hard line between them.

It's also worth noting that the definition you gave for Physical Bond was taken from an answer without a single upvote, which is an indicator that it isn't what the Chemistry SE community thinks (despite the fact that Google liked it very much and uses it in it's short-answer for the query). Another indicator that "Physical Bond" isn't really a type of bond is that there aren't definitions of it out there (Yahoo answers doesn't count), not even a Wikipedia article.

So why do we call them chemical bonds? 

That's pretty much because the bonds we call chemical are those which make up molecules, salts and other materials which are mainly studied by chemistry. The definition of chemical bond varies a lot (as explained here), but they are the bonds which create what chemists study. 

Other types of bonds which aren't called chemical are the intermolecular interactions, which are obviously of interest to chemistry, but also the Strong Force, which is the force that binds protons and neutrons (nucleons) together to form the nucleus of an atom. (I hope it's okay to call Strong Force a bond, I'm not sure to be honest).