The Higgs mechanism has been found to give mass to elementary particles only (short of neutrinos) [3]. This is important to note, as the mass of hadrons is far larger than the sum of their constituent elementary particles [4]. The rest, and vast majority, is found in the bound energy (eg the Strong Interaction) of the elementary particles (eg quarks) [1][2].

Regarding "Inertial mass", I want to note the following definition for clarity:

Inertial mass is a measure of an object's resistance to acceleration when a force is applied. [5]

References

D H. "Relation between binding energy and inertial mass". Physics Forums. Published: 2012-07-08T01:12 (Accessed: 2024-08-13T05:39Z). https://www.physicsforums.com/threads/relation-between-binding-energy-and-inertial-mass.619191/post-3987273.

It's not the Higgs field. The Higgs gives elementary particles their masses. The strong interaction gives protons and neutrons their masses. There is a whole lot of energy bound up in those protons and neutrons. That's why protons and neutrons are considerably more massive than than the sum of the masses of the quarks that form them.

"Why we can take for granted that energy binding is associated with inertial mass?". Davius. Physics StackExchange. Published: 2023-05-17T10:07:08Z (Accessed: 2024-08-13T05:44Z). https://physics.stackexchange.com/questions/764416/why-we-can-take-for-granted-that-energy-binding-is-associated-with-inertial-mass

QCD energy binding (associated to a "cloud" of gluons joining together the three quarks) is responsible for the 99% of the inertial mass of a proton.

SuperCiocia. "Which particles does the Higgs Field give mass to?". Physics StackExchange. Published: 2020-08-04T03:33:37Z (Accessed: 2024-08-13T06:08Z). https://physics.stackexchange.com/questions/571121/which-particles-does-the-higgs-field-give-mass-to.

the Higgs field is responsible for the masses of all the elementary particles (including the Higgs boson) short of neutrinos.

Viktor T. Toth. "How is the strong nuclear force responsible for over 90% of mass?". Quora. Published: 2021-05-02 (Accessed: 2024-08-13T06:26Z). https://www.quora.com/How-is-the-strong-nuclear-force-responsible-for-over-90-of-mass.

Actually, it’s closer to 99% when it comes to protons and neutrons. Only about 1% of their masses come from the up and down quarks; the rest is binding energy.

"Mass". Wikipedia. Accessed: 2024-08-13T06:34Z. https://en.wikipedia.org/wiki/Mass#Definitions

Higgs makes it hard to push something. Gravity makes it hard to lift something.

The Higgs interaction contributes to both. Mass due to the Higgs interaction is a component of inertial mass, given that the Higgs boson gives mass the elementary particles contained within [3]. Inertial mass is a measure of an objects inertia (ie its resistance to acceleration when a force is applied) [1][2].

Note that the concept of "lifting" only applies in a gravitational field when a force is able to be created by pushing off of a surface — the resistance to the "lift" being created by the objects weight. If one is in free-fall, for example, the effects of gravity are no longer apparent given that one has no reference to the fact that they are falling.

References

"Inertia". Wikipedia. Accessed: 2024-08-13T06:43Z. https://en.wikipedia.org/wiki/Inertia

Inertia is the tendency of objects in motion to stay in motion and objects at rest to stay at rest, unless a force causes its speed or direction to change.

"Mass". Wikipedia. Accessed: 2024-08-13T06:44Z. https://en.wikipedia.org/wiki/Mass#Definitions.

Inertial mass is a measure of an object's resistance to acceleration when a force is applied.

D H. "Relation between binding energy and inertial mass". Physics Forums. Published: 2012-07-08T01:12 (Accessed: 2024-08-13T06:46Z). https://www.physicsforums.com/threads/relation-between-binding-energy-and-inertial-mass.619191/post-3987273.

The Higgs gives elementary particles their masses.

the other is called gravitational mass (what we feel due to gravitational attraction between two masses).

I'm not sure exactly what you are trying to say here, but I suspect you are perhaps referring to the following excerpt(s) from Wikipedia:

Active gravitational mass is a measure of the strength of an object's gravitational flux [1]

Passive gravitational mass is a measure of the strength of an object's interaction with a gravitational field. [1]

I've personally never heard the term "gravitational mass" before, but it could be found from the above two statements, or more succinctly stated:

Gravitational mass is the "m" in F = GmM/r2 (Newton's law of universal gravitation) [2]

References

"Mass". Wikipedia. Accessed: 2024-08-13T06:54Z. https://en.wikipedia.org/wiki/Mass#Definitions.
starkeffect. "Can someone explain to me how inertial mass and gravitational mass are the same and what this means outside the classroom?". AskPhysics. Reddit. Published: 2023-09-24T22:59:20Z (Accessed: 2024-08-13T07:00Z). https://www.reddit.com/r/AskPhysics/comments/16rayhv/comment/k228dew/

They are usually the same so the distinction is usually ignored.

This statement is rather dubious. Simply put, there has simply not been found any empirical difference between inertial mass and gravitational mass. [1]

References

"Eötvös experiment". Wikipedia. Accessed: 2024-08-13T07:07Z. https://en.wikipedia.org/wiki/E%C3%B6tv%C3%B6s_experiment