Do the Broglie's equation or the Higg's field could give "mass" to a photon?

I know that there are probably hundreds of questions about the mass of a photon, but none of those questions was able to clarify me some problems.

I was talking with a guy on Facebook that says that photons do have mass. He uses a term "maupertusian mass" (the term used is "massa maupretusiana" cause the discussion was made in portuguese) and it is attributed to Paul Langevin, but I found nothing about that. And he said that Poincaré somehow proved that that the photon do has mass and a NASA scientist found that it should be in the order of $$4\times10^{-44}$$ grams.

The problem is that I can't find any consensus about the definition of mass itself. For now, I will define mass as being the rest mass, if you could somehow put the particle in a scale and find a non zero value. I was told that photons do not have rest mass, but instead they have relativistic mass and it is that mass that give them their momentum. Then if I put a proton, for example, in a scale I would find its mass, but I would find nothing for the photon.

The most counter-intuitive thing is that photons do have momentum but they have no mass.

Most people think that momentum has to do with mass due to the relation $$p=mv$$, but if so, the photon could not have any momentum at all.

The explanation most people give (and I think that it is the correct one) is that all momentum of a photon is relativistic.

The "full equation" of energy in relativity is $$E^2=m^2c^4+p^2c^2$$. Since a photon has no mass, it has only momentum and it is related by its energy by $$E=pc$$.

That's fine. Correct me if I'm wrong: this momentum has nothing to do with actual mass. I mean, photons do not have rest mass and it implies that this momentum is only related to energy but nothing to do with mass.

The problem arises when we take in account de Broglie's equation.

$$\lambda=\frac{h}{p}=\frac{h}{mv}$$

This equations works for the photon cause it has momentum. But does it implies that a photon have mass? We can simple use $$v=c$$ and find the mass of a photon. Is that correct? If this momentum is not due to actual mass but it is relativistic instead, then there is no sense to find a mass for a photon using this equation.

I simply can't grasp this thing because I can't even define what exactly mass is to begin with! And to be worse, relativistic mass is an obsolete term in many textbooks. The same guy said that it is possible to a photon to interact with the Higg's field and gain mass. I can't grasp it either.

To summarize, can someone please explain to me the following questions? When we normally talk about mass, what exactly we are talking about(e.g. mass of a proton)? Does photons really can't have mass? The term "maupertusian" or similar was used in some scientific literature? It is true to say that momentum of the photon has nothing to do with actual mass and then there is no sense to use de Broglie's equation to calculate it? Can photons interact with the Higgs field? And finally, is there a possibility that a photon could have mass in some way or in some circumstances? I mean, the same way a proton has mass, for example? Or not, it is utterly impossible to even find a value for the mass of a proton no matter the circumstance?

I hope someone can understand my questions and could help me to clarify it. Sorry if I ask too much, but there some thing that I really searched for answers but none solved the problem. Many thanks in advance!

Photons have energy and momentum but no mass. In modern physics, “mass” means “Lorentz-invariant mass”, defined by $$m^2c^4=E^2-p^2c^2$$. Your deBroglie equation with $$m$$ is non-relativistic and doesn’t apply to photons.

You are supposed to ask only one question at a time (not twelve!) so I have answered a few of your most important ones.

I think that some of your questions are already answered, so I would like to clarify a few things left over. Any object that travels at the speed of light cannot have any mass at all. So under no circumstances can a photon have mass. Relativistic mass does not exist. It is just a term invented to make the corrected equations of mechanics look similar to Newton's. (Newton defined momentum as $$p = mv$$, while Special Relativity defines it as $$p = \gamma mv$$ so $$\gamma m$$ was 'called' relativistic mass, which changes with speed)
SIDE NOTE: The momentum of a photon has no relation with it's mass. The photon of a light is defined as $$p = \frac{E}{c}$$ and so using this we get: $$\lambda = \frac{h}{p} = \frac{\frac{h}{E}}{c} = \frac{hc}{E}$$