In general you cannot determine this information from the public data feeds- the purpose of Iceberg orders in particular is to be hidden and difficult to detect. Also, there isn't really a standardized definition of an Iceberg, so there would be no consistent way of performing this labeling. If you could reliably identify them though, there is good money in that :). You could get an approximation of market order identification by combining trades in a given symbol that have identical timestamps- however that will treat both market orders and deep-book limit orders the same way.
In regard to the updated question- depth of book data is much more extensive than what is provided at the data vendors you mentioned. The most common type of available market data is so-called TAQ (trade and quote) or NBBO.
Generally, providers of this so-called Level 1 or L1 data will offer all trades, as well as the best available bid and ask price, along with the volume available at that price. In the case of Polygon, the data comes from the consolidated feed which means that it includes trades from all public exchanges, as well as the best bid and ask from all exchanges.
Depth of book data, on the other hand, would individually show every order which is currently resting on the order book for each exchange. Generally to work with depth of book data, you would need to reconstruct the order book yourself and "emulate" the matching engine to figure out what the current best bid or ask are. Depth of book data tends to be harder to work with and is much bigger, but on the other hand it opens up new kinds of analysis that you can't do with L1 data, such as analyzing the structure of the order book and the composition of individual orders in it.
There are compromise products, so-called Level 2 or L2 data, which will already have the order book creation done, and will give you the aggregate volume at each level of the book. You can see this, for example, in the crypto feed at polygon.io, or the L2 proprietary feeds offered by most exchanges.
Currently, there is no standard consolidated feed for either L2 or DOB data, and generally users who want it will need to get the data themselves and perform the consolidation suitable to their own purposes. Right now there is a proposal that is working it's way through the SEC to make L2 data more easily available, but the state of affairs now is that it's still quite expensive for an individual to work with.