Let's proceed stepwise as instructed, prioritizing quantitative data typically found in physicochemical databases, with a focus on strong bases (since all your candidates are organic superbases).

### Step 1: Identify Key Physicochemical Properties

For the classification of organic bases, the **most critical physicochemical property is pKa (of the conjugate acid)**, as it quantitatively reflects basicity. 

- **pKa of Conjugate Acid (pKₐH):** A higher pKₐH means a stronger base.
    - For organic superbases, **pKaH > 13–15** is typical.
- **Other properties** possibly relevant: number of basic sites, H-bond acceptor count, steric hindrance (often as "cone angle"), but pKaH is primary and quantifiable.

**Relevant database values:**
- pKₐH in water (and sometimes DMSO/acetonitrile for insoluble substances)
- Number and nature of basic nitrogen atoms or phosphorus centers

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### Step 2: Compile Quantitative Data for Each Substance

1. **1-methyl-2,3,4,6,7,8-hexahydropyrimido[1,2-a]pyrimidine**  
   - This is a bicyclic amidine derivative (no phosphorus).
   - **Literature pKₐH (water):** ~13–14 (Amidine-type bases)  
   - **Quantitative indicator:** 2 nitrogen atoms with high electron density, H-bond acceptors = 2.

2. **2-tert-butyl-1,1,3,3-tetramethylguanidine (tBu-TMG)**
   - A guanidine-type superbase; widely used standard.
   - **Reported pKₐH (water):** ~13.6–13.9  
   - **H-bond acceptor count:** 3; **Basic nitrogen = 1 central strongly basic; 2 flanking less basic**

3. **N-[dimethylamino-ethylimino-[[tris(dimethylamino)-lambda5-phosphanylidene]amino]-lambda5-phosphanyl]-N-methylmethanamine**  
   - This is **P₁-t-Bu** (aka Schwesinger base, Verstatile base BEMP, a phosphazene base).
   - **Reported pKₐH (acetonitrile):** ~25–28 (much higher in less protic solvent, as water may decompose it)  
   - **In water:** Not stable, but **by acetonitrile standards, orders of magnitude more basic**
   - **Number of basic sites:** Nitrogen and phosphorus center act together.

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### Step 3: Classification According to Quantitative Data

#### **A. By pKₐH (Quantitative Basicity Strength)**

Since **pKₐH is a strictly quantitative, database-sourced property**, let's set conventional groupings:

- **Group 1: "Strong Organic Bases" (pKₐH 12–15)**
    - e.g. amidine, guanidines, weaker phosphazenes

- **Group 2: "Superbases" (pKₐH > 20 in acetonitrile, or much stronger than guanidines/amidines)**
    - Not achievable in water for all; compare in acetonitrile if necessary.

#### **Applying the Data:**

| Substance                                                                               | pKₐH (water/acetonitrile) | Group Based on Quant Data              |
|-----------------------------------------------------------------------------------------|--------------------------|----------------------------------------|
| 1-methyl-2,3,4,6,7,8-hexahydropyrimido[1,2-a]pyrimidine                                | ~13–14                   | Group 1: Strong Organic Base           |
| 2-tert-butyl-1,1,3,3-tetramethylguanidine                                              | ~13.6–13.9               | Group 1: Strong Organic Base           |
| N-[dimethylamino-ethylimino-[[tris(dimethylamino)-λ5-phosphanylidene]amino]-λ5-phosphanyl]-N-methylmethanamine | ~25–28 (MeCN)            | Group 2: Superbase (Phosphazene)       |  

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### Step 4: Justification and Ambiguity

#### - **Clear Justification**:
- The first two substances have pKₐH values around 13–14, firmly in the "strong base" range.
- The Schwesinger phosphazene base has an acetonitrile pKₐH near 28, **far exceeding di- and tri-alkylguanidines**; by any quantitative measure, it belongs in a more basic group.

#### - **Ambiguity?**
- The solvent effect must be considered; but the difference between pKₐH ~14 and ~28 is **over 10⁸ in proton affinity** in polar aprotic media, supporting a split into two distinct quantitative categories.


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## **Final Classification (by Quantitative Basicity, pKₐH):**

### **Group 1: "Strong Organic Bases" (pKₐH 12–15 in water)**
- **1-methyl-2,3,4,6,7,8-hexahydropyrimido[1,2-a]pyrimidine**
- **2-tert-butyl-1,1,3,3-tetramethylguanidine**

### **Group 2: "Superbases / Phosphazene Bases" (pKₐH >20 in MeCN)**
- **N-[dimethylamino-ethylimino-[[tris(dimethylamino)-λ5-phosphanylidene]amino]-λ5-phosphanyl]-N-methylmethanamine**

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### **Summary Table** (for clarity)

| Group        | Criteria (Quantitative)                      | Substances                                                     |
|--------------|---------------------------------------------|----------------------------------------------------------------|
| Group 1      | pKₐH = 12–15 (water)                        | 1-methyl-..., 2-tert-butyl-1,1,3,3-tetramethylguanidine        |
| Group 2      | pKₐH > 20 (acetonitrile, superbase class)   | N-[dimethylamino-ethylimino-...phosphanyl]-N-methylmethanamine |

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### **Conclusion**
Placement into two strictly quantitative groups is justified **by measured pKₐH values**, not by structural intuition alone. The cutoffs and assignments use data from standard chemical property databases and the chemical literature. No further groups (or "all-in-one" group) are justified based on available quantitative physicochemical data.