Learn English – Are these two sentences gramatically correct?

It seems highly awkward and clumsy to me; I would say no, you cannot.

The definite article has many uses. You seem to be wanting to leave it out here because of the half-truth often stated that a definite article requires that the noun phrase in question be already within the scope of the discourse. This is not entirely false, but it only a part of the whole truth.

Another function of the definite article is to signal that the noun phrase is somehow modified or more narrowly defined than the bare noun (phrase) itself.

Thus, for example, it is perfectly natural that the second paragraph in this answer starts with “the definite article” since the definite article has already been introduced in your question and is within the scope of our current (virtual) discourse. In the next line, it is also natural that I use “the noun phrase” because it is modified and narrowed down by “in question”. If this narrowing down had not been present, the sentence would have been awkward:

… a definite article requires that the noun phrase be already within the scope of the discourse

– would not work in this context since “noun phrase” is not something that exists in the scope of the discourse.

In your example, “on my phone” narrows down the head of the noun phrase, “pictures”, which means that it is perfectly natural to use a definite article. You are in fact referring to a specific, definite subset of pictures, namely the ones that are located on your phone.

Leaving out the definite article is fine in the sentence itself—as stated in the thread you link to, it simply forces “pictures” to refer to an indefinite, undefine set of pictures, which is then taken to be just any possible pictures and ends up meaning that you love having pictures on your phone. An undefined, vague notion of “pictures” is introduced into the scope of the discourse, but it remains to some degree ‘unreferenced’, as it were.

There is nothing wrong with this, but since the next sentence uses “the pictures” without narrowing down the set of pictures, this instance of the definite article does require that a specific, defined set of “pictures” be available in the discourse—and no such set is found. Only a looser, undefined set of “pictures” is found, which is not enough.

When we think of "additional" or "extra" parts of the sentence like this we are thinking about functions not parts of speech or types of phrase.

The proper terms for this type of function is ADJUNCT. An adjunct is part of a sentence that is not necessary for the grammar. This means it is not necessary for the sentence to be grammatical or make sense.

The subject and object are COMPLEMENTS of the verb. Certain verbs set up slots for both subjects and objects; some for just subjects; some for subjects, objects and other complements. If part of a sentence is filling one of these slots, it is a complement of some description.

However, the sections in bold in the Original Poster's question:

• to the finance department
• at the restaurant
• with Mr. Barber

... do not fill any of these slots. They are entirely extraneous in terms of the structure of the sentence. The sentences are perfectly grammatical without them:

• He was giving a presentation.
• She was having lunch.
• They were going to a meeting.

It is this property of not filling a particular slot in a sentence or phrase which makes an adjunct an adjunct.

One last thing to mention is this. Often, when we talk about adjuncts, we are talking about the structure of the sentence in terms of immediate constituents of the verb phrase. However, strictly speaking, adjuncts can occur in any phrase. A good illustration is the Original Poster's third example:

• They were going to a meeting with Mr. Barber.

Now if Mr Barber was going to the meeting with them, we would regard this as an adjunct in the verb phrase - in other words as a general adjunct of the sentence.

However, if they are travelling to the meeting without Mr Barber, but Mr Barber was at the meeting, then with Mr Barber is an adjunct in the noun phrase 'a meeting with Mr Barber'. Here the preposition phrase with Mr Barber is modifying the noun meeting, not the verb phrase were going to a meeting. We therefore say that with Mr Barber is an adjunct in the noun phrase. It is not filling any special slot in the phrase. Notice that this noun phrase, a meeting, is well-formed without the preposition phrase.

One last addendum: some people refer to adjuncts as ADVERBIALS. However, this is a bad term as it has associations with the word adverb. Now adverb is a part of speech, not a function in a sentence. Generally speaking, writers who use the term 'adverbial' are generally unable to distinguish whether they are talking about parts of speech, types of phrase or functions. The terms is confusing and those who use it are - more often than not - unwittingly confused! It is a term that should be banished from any serious discussion of English grammar.