Summary
The systematic chemical names of many important biological molecules are too long to write routinely in full, and in any case were preceded historically by abbreviated forms which generally stressed the distinguishing features of importance to those studying their metabolism. Thus, as the poster points out, a molecule such as dATP (image below) has a base-(deoxy)sugar-phosphate(s) structure, and can be termed a deoxynucleotide. Adding triphosphate provides the additional information that there is more than one phosphate. A base-sugar(deoxyribose) combination is, as the poster states, a deoxynucleoside, but by extending this name to deoxyribonucleoside triphosphate, one indicates the addition of phosphates, so that one is, by definition, describing a deoxynucleotide. The reference to ribonucleoside indicates that the sugar is ribose, a specification lacking from the other definition. However, in practice these two terms are used to describe the same entity, one which might a little more completely describe as 2′-deoxyribonucleoside triphosphate (to indicate the position of the reductive change). This answer attempts to explain with precision the aspects of the nomenclature that are likely to be important to the biochemist or molecular biologist.
Disclaimer
The original question might be considered not to represent a biological problem, and one might argue for closing it on that account. However because of the very nature of trivial nomenclature, it is difficult for those without experience of the usage in a field to find the answer. In answering it I have tried to broaden the scope to include the aspects of structure that I feel are most likely to be to biological students.
Answer
It is convenient to start with ribonucleoside triphosphates (rather a deoxy- ribonucleoside triphosphates), and I have chosen a specific example in the molecule that is generally known as ATP:

Alternative names for this ribonucleoside triphosphate, in decreasing order of complexity, include:
- [[(2R,3S,4R,5R)-5-(6-aminopurin-9-yl)-3,4-dihydroxyoxolan-2-yl]methoxy-hydroxyphosphoryl] phosphono hydrogen phosphate
- 9-β-D-ribofuranosyl adenine 5′-triphosphate
- adenosine 5′-triphosphate
- adenosine triphosphate
- ATP
Of interest to us are the three components of the molecule: a purine base (adenine), a sugar (ribose), and three linked phosphates. This is not reflected in the chemical nomenclature (1), so let us focus on the, albeit rarely used, biochemical nomenclature (2).
- The ‘9’ specifies the position of the adenine ring that is linked to the sugar. This is not of particular interest in the context of the question. (The numbering of the ring is, however, relevant to describing differences between or modifications of the purine and pyrimidine bases1.)
- The D-ribofuranosyl is a form of ribose (see below). The ‘D’ indicates that it is D-ribose enantiomer rather than the alternative, L-ribose. However, as all natural sugars are ‘D’ this is generally omitted (although discussions of why all natural sugars are ‘D’ abound2).
- Ribofuranosyl indicates that the ribose is in the five-membered hemiacetal ring form (rather than the straight chain or the six-membered pyranose ring), but this is always the case in nucleosides, and so is not of particular interest.
- The 5′ specifies the position of the sugar ring to which the phosphates are attached, the prime (′) indicating that the numbering is for the sugar ring (not that of the base). This is important, as it is relevant to the linkage in the (asymmetric) phosphodiester bond made to the 3′-OH:

The image above is for DNA, but before addressing the nomenclature there, we would point out:
- The common α, β, γ designation of the three bond phospho-ester bonds. This is very relevant to the formation of polynucleotides (DNA and RNA) from their substrates, and in general to the utilization of the free energy the hydrolysis of these bonds for other processes3.
Finally we come to a structure of interest to the poster, dATP:

- In the designation, 2′-deoxyadenosine 5′-triphosphate, the 2′-deoxy indicates that it is the 2 position in the ribose ring (′) that has had the oxygen removed (deoxy), i.e. OH is reduced to H. Although common knowledge, it is important that the student new to the subject understands this, as if it were the 3′ position that was reduced, it could not form the phosphodiester bonds of DNA. And it is relevant in many other contexts. For example the chain-termination method of DNA sequencing uses 2′,3′-dideoxy nucleotides to inhibit such bonds forming4, the single difference between ribose and deoxyribose allows us to understand the profound affect on the susceptibility of RNA to alkaline hydrolysis in nature5, and certain RNA molecules are hydroxy-methylated at this position.
Related information in SE Biology answers
1 Topical example related to base modification
2 Speculation on the origin of biological homochirality — not SE Biology
3 Why does ATP sometimes produce ADP and sometimes AMP?
4 Sanger sequencing by chain termination
5 Difference in stability of DNA and RNA