As long as the secondary is left open, a transformer primary behaves exactly as an inductor, since the secondary has substantially no effect on the circuit.
The reason behind this behavior is that the primary and secondary are coupled by the magnetic flux that flows trough the transformer core. The magnetic flux depends on the current flowing in a transformer winding. Therefore, if in the secondary doesn't flow any current, no flux is generated by the secondary that can interact with the flux generated by current in the primary, hence the secondary doesn't interact with the primary whatsoever.
Keep in mind, though, that even with the secondary open, a transformer core is usually made with ferromagnetic materials (or ferrimagnetic materials, in case of ferrite cores), which is highly non-linear. This means that the inductance of the primary is not constant (as that of an air-wound coil), but depends on the amplitude of the current in the winding.
Therefore you cannot simply substitute the transformer primary with an air-wound coil with the same inductance and internal resistance and expect identical behavior (always assuming the secondary is open). To have identical behavior you should replace the transformer primary with a coil wound on a core made of the same material with which the transformer core is made (and even the same shape).
EDIT (prompted by a comment - integrating information I provided in comments)
So you are puzzled by a theoretical primary inductance value that is about 5 times higher than what you measure. In reality this is not surprising, for a variety of factors I'll try to explain below.
You should understand that non-linear inductors (NLI in the following) are nasty beasts, and a cored transformer is just a NLI when you leave the secondary open.
The main problem for a NLI is that the very concept of inductance is not well-defined, if for inductance you intend a constant parameter that expresses the relationship between current and magnetic flux. Even if you treat L as function of the current, instead, you are not getting the right picture, it does account for the non-linearity (saturation), but it doesn't account for the memory effect due to residual magnetization in the core, i.e. the effect that causes hysteresis in the core.
The fact that a NLI is so complicated also implies that your calculations are somewhat wrong in theory: they assume the classical LR circuit where L is a constant and the step response is an exponential decay. In reality, with a NLI, the curve you see on the scope is not a true exponential, but only something that resembles it. Therefore you shouldn't expect that the rule you apply to infer L (i.e. that the decay brings the curve under 1% level after 5 time constants) is perfectly valid.
This is a first source of error. So your computed value is affected by an error just before you measure the real thing (call it a model approximation error, if you want).
Then there are the differences in measurement techniques: the "inductance" of a NLI is influenced by many parameters of the signal used to probe the inductor, especially amplitude and frequency. Hence the shape of the signal may also alter the results. As you have seen even two different LCR meters give results that differ by a factor of 3 or 4.
I won't enter in further details, but I'll point you to this application note of Keysight (former Agilent, former HP): Impedance Measurement Handbook
. The following are relevant excerpts:
5.2 Inductor measurement
5.2.1 Paracitics of an inductor
An inductor consists of wire wound around a core and is characterized by the core material used. Air is the simplest
core material for making inductors, but for volumetric efficiency of the inductor, magnetic materials such as iron,
permalloy, and ferrites are commonly used. A typical equivalent circuit for an inductor is shown in Figure 5-9 (a). In
this figure, Rp represents the magnetic loss (which is called iron loss) of the inductor core, and Rs represents the
copper loss (resistance) of the wire. C is the distributed capacitance between the turns of wire. For small inductors
the equivalent circuit shown in Figure 5-9 (b) can be used. This is because the value of L is small and the stray
capacitance between the lead wires (or between the electrodes) becomes a significant factor.
5.2.2 Causes of measurement discrepancies for inductors
Inductance measurement sometimes gives different results when a DUT is measured using different instruments.
There are some factors of measurement discrepancies as described below:
Test signal current
Inductors with a magnetic core exhibit a test signal current dependency due to the nonlinear magnetization
characteristics of the core material as shown in Figure 5-13 (a). The level of test signal current depends on the
impedance measurement instrument because many of the instruments output a voltage-driven test signal. Even
when two different instruments are set to output the same test signal (OSC) voltage, their output currents are
different if their source resistance, Rs, is not the same as shown in Figure 5-13 (b).
To avoid the measurement discrepancies, the OSC level should be adjusted for a defined test current by using the
auto level control (ALC) function or by determining the appropriate test voltage setting from the equation shown in
Figure 5-13 (b).
5.3 Transformer measurement
A transformer is one end-product of an inductor so, the measurement techniques are the same as those used for
inductor measurement. Figure 5-18 shows a schematic with the key measurement parameters of a transformer. This
section describes how to measure these parameters, including L, C, R, and M.
So, in the end, you cannot expect to get a result which is extremely precise, especially with a non-controlled test setup. If you get a value which is of the same order of magnitude of the real value (i.e. within an 1:10 ratio) you should be happy. Then yes, the values you are getting are perfectly in line given the test setup you have described.