In my estimation, you are asking two different questions here. The one about what might tell your students is easier to answer, but there are lots of different answers that are reasonable. Here is what I would say to, e.g., a college student in any course before linear algebra:
A function is a machine that takes in numbers and spits out other numbers. A formula is a mathematical expression that I can use to calculate. Functions are more general objects than formulas, because not every function can be written as a formula. My favorite example is the floor function, which works like this: it takes in a number, cuts off the decimal, and spits out the whole number that's left over. [examples]. This function makes perfect sense, even though I can't write down a "formula" for it like x^2 or $\sin(x)$.
None of the first three sentences are true (the fourth one is!), even to the extent that they are meaningful. But the lies are mild, and they help simplify the language to get the point across.
The second question is what the difference actually is. I will not come to any forceful conclusion on this question, but instead frame my thoughts as a series of remarks.
Remark 1. This question has been discussed a lot on MSE, because, as you have observed, it is subtle, and it concerns relatively elementary objects. Exactly which question it is a duplicate of is not totally clear, though.
Remark 2. I can't open the first link, but based on the language used, it kind of sounds like this source is using the programming definition of a 'function', which differs in a subtle way from the mathematical definition. In particular, we usually do not agree with [c]: for us, the function is the encapsulating object, and there are various methods by which a function's values may be given, one of which is a formula (see this Wikipedia quote)
Remark 3. Continuing on this, you write
We know a function can be used to calculate to find the result of the calculation from the range with the given number taken from the domain [a]...
I know no such thing. A function may encode a calculation (or it may not!) but it is odd to me to say that we use a function to calculate elements in its range. For instance, in your example I would not say I am using the function $P$ to calculate $P(3)$; I guess I can see why you would say this, but it doesn't reflect how the calculation actually proceeds. When I am actually getting around to doing the computation, what I am using is the expression $P_0e^{kt}$ by substituting $t=3$. The result of this calculation is certainly in the range of $P$; this is guaranteed by the equation $P(t)=P_0e^{kt}$... indeed this equation is what convinced me to use the expression to compute $P(3)$.
Remark 4. The word "formula" did not appear in the previous remark, what's up with that? Well, in casual speech, I would describe both $P(t)=P_0e^{kt}$ and $P_0e^{kt}$ as formulas for $P(t)$. If I am required to be more careful, I would not use the word at all.
Remark 5. Logicians have strong opinions about [read: a definition of] what a "formula" is, that does not align well with the common-language meaning, despite being related. (The tension is well-illustrated by this answer.) To answer your question, the three equalities in your question undoubtedly are formulas if we are using the logician's definition. Still, I would not call them formulas because for me a formula needs to be something that I can compute with; those equalities look to me like a "record" of a computation that has already been completed.
As this answer has now been accepted, let me highlight
Mauro Allegranza's very useful comment [emphasis mine]:
The simplest explanation is in the answer to the linked post: "A formula is a string of symbols, arranged according to mathematical grammar [i.e. an expression of the mathematical language]. A function is a mathematical object." The clear distinction is that between the world of (mathematical) objects and the language used to speak of them (exactly like natural language). –
Best Answer
It's useful to think of it this way: A processor's speed is inversely proportional to the time it takes to complete the task. Therefore, for processor A we can give a "non-dimensional speed" of $\frac{1}{42.5}$. For processor B, this value is $\frac{1}{32.9}$. It's easy to see that B is faster in this case, because the speed is greater.
Now you want to compare their speeds, or in particular answer the question "How many per cent is processor B faster than A?". So you can just calculate the ratio of their speeds like this: $$ \frac{\left( \frac{1}{32.9}\right)}{\left( \frac{1}{42.5}\right)} = \frac{42.5}{32.9}\approx 1.2917 $$ Therefore, our answer is: Processor B is faster than processor A by $29.2~\%$.
Should you calculate it the other way around, you'd be asking the question "How much slower is processor A compared to processor B?".