Many verbs that take subordinate clauses can take clauses that look like main clauses.
- Ich glaube, dass der Termin verschoben wurde.
- Ich glaube, der Termin wurde verschoben.
This creates a terminological problem. Nebensatz usually refers to clauses that are subordinate and verb-final, just as in the first sentence above. Hauptsatz, on the other hand, usually refers to main clauses (i.e. not subordinate, but independent) that, in the case of declarative clauses, are verb-second.
In the second sentence above, we have a subordinate clause with "main clause word order", i.e. verb-second. Grammarians have come up with the term uneingeleiteter Nebensatz, with uneingeleitet referring to the missing conjunction.
Although the two sentences above are equivalent in meaning, this is not always the case. In fact, for your example, I find
#Ich denke, dass ich die nehme.
close to unacceptable with the intended meaning of communicating an order to a waiter. (The sentence is fine when communicating to someone else which pizza you intend to order.)
One assumption that would explain this discrepancy is this: When the content of the matrix clause is sufficiently "light" or close to redundant (after all, opinions or beliefs can be expressed without prefixing ich denke or ich glaube), the illocutionary force of the sentence is carried by the embedded clause when that clause is able to do so, i.e. when it has the form of a main clause (verb-second).
In other words,
Ich denke, die nehme ich.
is equivalent to
Die nehme ich!
Ich denke, dass ich die nehme.
Ich gedenke, diese Pizza zu nehmen.
Ich spiele mit dem Gedanken, diese Pizza zu nehmen.
in so far as that it can't be used to relay an order to a waiter.
For the terminus technicus of illocution see Wikipedia.