What is the difference between conditional and inducible activation?
In my opinion the terms inducible and conditional are synonyms. Together with the term constitutive they refer to how transcription takes place, that is, the second phase in the central dogma that states that genetic information is propagated from replication to transcription to translation. The proteins that are produced during translation by ribosomes can have a feedback effect on replication and transcription.

When proteins bind to the promoter region of genes, they can induce or inhibit transcription. If a gene is preceded by a promoter region that always promotes transcription, it is said that the gene has a constitutive promoter. If the gene is transcribed conditionally, it may refer to proteins having to bind to the promoter region so that transcription can take place. There could be inverse situations and I believe the Cre/Lox technology is an example of such a situation.

Take the following as an example. A gene could be lethal meaning that if the protein it encodes is expressed it will kill the cell. It could have a promoter that promotes expression if a protein with an inhibitory effect is not bound to the promoter. Suppose I have these cells in a culture plate and I add some compound to the culture plate. The compound diffuses into the cells and binds to the inhibitory protein that is bound to the promoter region. With this binding of the compound to the protein that is in turn bound to the promoter region, the protein loses its ability to bind to the promoter region and inhibition of transcription is uplifted. Transcription then continues and expression of the lethal protein takes place after which the cell dies.

This could be used in practice as follow: if you were selling seeds to farmers, you would want them to buy seeds next year again, so you want the plant not to be able to produce viable seeds. 

Coming back to your question, if you want to know more about the mice in question, you should contact the breeder as nomenclatures differ. See this article on interpreting mouse genetic nomenclature:

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3039125/