I think you're getting too hung up on the titles.  Look for someone offering a job with zero experience.  That's it.  Not trying to be a dick, but you don't have any experience (that they're asking for). 

You also shouldn't assume that all internships are "for students and not Master's graduates".  Some internships are specifically for students, but until just 3 or 4 years ago "intern" was the official term used in the US by NCARB and AIA for any unlicensed person working toward completing IDP/AXP experience, so most recent grads with 0-3 years of experience - and sometimes many more years than that - were titled "interns".  A lot of firms are still using that terminology, so if a firm is advertising for an "intern" you should try to get more information:  if it's specifically for students only, and particularly if it's unpaid and for academic credit, then of course that's not going to work for you - but if what they mean is just an entry-level position and/or a short-term position, then by all means apply for that.

To avoid the issue altogether apply unsolicited with a solid portfolio and good resume, there are plenty of jobs out there.

Finding a job when fresh out of school is the equivalent of a full time gig, maybe 1 and half.  If you're not working 8hr per day on your applications and researching offices in your area, then you're doing it wrong.

Also since you're foreign, most US offices don't want the added cost to sponsor visas which limits the amount of jobs. Can you work in the country where you got your degree? Some EU countries give you a year to find work without a visa. 

In architecture in the US "junior" vs. "intern" really are interchangeable, and both just mean relatively entry-level.  One is not a step up from the other.  As Formerly noted above, "Intern" in architecture was used for the last 70 years to mean people working toward licensing. It's not like in other professions where an intern is almost always a student.  In architecture an intern has meant someone with a B.Arch or M.Arch who is somewhere in their years between graduating and completing their license exams.  That period can be as short as 3 years but averages more than 6 years.

It was very recently (2017) that NCARB discontinued using "intern" and recommended replacing it with "architectural associate" or "design professional" - but both of those new suggested titles actually violate laws in a lot of states, and a lot of firms already use "associate" in different ways to mean more senior people, so these new titles just haven't really caught on well to replace "intern".  That's why there are still a lot of job postings for "interns", and also why "junior" is in more frequent use than it used to be.  You really should just apply to anyplace that appeals to you, and sort out the titles later.

I have seen a lot of firms hiring people from Europe on J-1 visa's quite frequently. It gives them time to assess whether a person who seemed talented in an interview actually has the chops to contribute holistically as a team member in the long term at a relatively low cost. Then they are fairly comfortable to invest in visa fee and so forth as they are far more assured about what value you bring to them.

As other's said, I wouldn't get hung up on titles. The term "intern" is an old NCARB terminology. At this point of time, graphic capability is likely to be the only relevant skill you might have acquired so far based on your background. There's a fair bit of training ahead to get a firmer grasp/start from scratch to learn assemblies, local codes, co-ordinating with consultants, local authorities, etc based on project-typology and firm size/culture and role. If I were you, I would really narrow down what kind of work excites you and apply with a portfolio showing your graphic chops and enthusiasm about learning with their team. I also wouldn't be too rigid with salary expectations unless you have tons of competing exciting offers. It's more important to have the opportunity to grow in a healthy environment than just go for a few thousand dollars more on an entry-level gig.

I would also add that all job-descriptions that excite you requiring 0-5 yrs experience are fair game. I've seen people, who after working just one year developing the skills to co-ordinate and produce comprehensive sets for fairly complex custom projects, compared to some average PA's having even 10yrs of experience. It's all about getting the right opportunities and  making the most of them to build up  value in yourself. 

My personal experience was that those first couple of jobs were all about developing skills that were relatable so I could get the better jobs down the road.  Employers want bankable skills that make them money.  I have looked at a ton of great graphics in many recent grads portfolios, but until I see any working drawing experience even a small house  / renovation, I'm not interested.  This is critical for small offices where staff need to do everything ( and a great learning environment in my opinion ) but in a large office it can be easier to hide and not get into the meat of a project, or get stuck doing more minimal tasks on a project.  Large office environments certainly have great appeal and nice to put on the resume but the small office environment is where you can learn more skills for career development.

I think the visa situation might get better once you move over here.

I suspect that most of the trouble you're having is because employers are unsure about whether they can legally hire you.  J-1 is a type of student visa - it does allow for up to 18 months of training after graduation, but usually only if the student receives a job offer before graduation.  If you know for a fact that there's some exception to that rule, that will allow you to work in the US for 18 months even though you've already graduated, then you should probably spell that out (as simply as possible) very early in your cover letter.  Unfortunately the job of educating your prospective employers on this is going to fall to you, because that's not a common visa situation that they're going to understand. 

Also it's not clear from your posts so far whether you're already in the US?  If so, you should make that very clear, and plainly state that you're readily available to interview in person.  I get so many emails from students and recent grads in other countries - typically with nothing even suggesting that they've done any research on our firm, and usually assuming that they can interview by Skype or that we'll hire them without any interview at all.  Do you have a local address on your resume?  If not it might be worth getting a mail forwarding service, just for the sake of having a New York address.  If it appears that you're not local then employees are much less likely to even review your application, because of the hassles of interviewing, worries about how soon you could relocate, and unknowns about the visa process. 

Also if you have any work at all that demonstrates basic familiarity with the types of construction that the firm does, it's important to convey that.  Hiring an entry-level person always involves a significant time commitment to training - and with an international applicant it can be a lot more time-consuming. The more evidence you can provide to show of skills that can be immediately useful in some way, the more likely someone is to think that the training commitment can be balanced out.

The visa issue is a little complicated.  There are a few different J-1 categories - it sounds like you're trying to use the "trainee" category, not a student category, but the trainee category requires a minimum of a year of employment in your field in your home country after you graduate, before you're eligible for J-1 training in the US, so are you sure you'll qualify, since you seem to be saying you don't have any employment experience?  Also the US position is supposed to exist only for training purposes - it's not allowed to be a job that would ordinarily be filled by an employee, so we aren't allowed to publish an ad for a employee position but then fill that with a J-1 person - but it sounds like those are the kinds of ads that you're applying to, so no wonder nobody's responding.  I think instead of applying to published employment ads you should apply directly to firms that interest you and explain to them that you're looking for a J-1 training opportunity, and then explain how that works.  Then an interested firm can decide to create a training setting for you, and not get in trouble for filling a regular job with a J-1 trainee.