Establish distinct indexing state in UI for the new UI

[Bernardin-MS]

The Tag Parser Initializing message is still being reported as Indexing with Indexing being reported as Parsing:

I thought that was going to be fixed?

Originally posted by @sean-mcmanus in #10126 (comment)

[Bernardin-MS]

There is currently no distinction between indexing or not in the updateStatus -> client.ts. This needs to be added in order for the new UI to correctly display when indexing is being done.